Physics for Scientists and Engineers with Modern Physics

Douglas C. Giancoli

Chapter 9

Linear Momentum - all with Video Answers

Educators

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Chapter Questions

Problem 1

(1) Calculate the force exerted on a rocket when the propelling gases are being expelled at a rate of 1300 $\mathrm{kg} / \mathrm{s}$ with a speed of $4.5 \times 10^{4} \mathrm{m} / \mathrm{s}$ .

Bruce Edelman

Bruce Edelman

Numerade Educator

Problem 2

(I) A constant friction force of 25 $\mathrm{N}$ acts on a 65 -kg skier for 15. S. What is the skier's change in velocity?

Farhanul Hasan

Numerade Educator

Problem 3

(II) The momentum of a particle, in SI units, is given by $\vec{\mathbf{p}}=$ $4.8 t^{2} \hat{\mathbf{i}}-8.0 \hat{\mathbf{j}}-8.9 t \hat{\mathbf{k}}$ . What is the force as a function of time?

Bruce Edelman

Numerade Educator

Problem 4

(II) The force on a particle of mass $m$ is given by $\vec{\mathbf{F}}=26 \hat{\mathbf{i}}-12 t^{2} \hat{\mathbf{j}}$ where $F$ is in $\mathrm{N}$ and $t$ in s. What will be the change in the particle's momentum between $t=1.0 \mathrm{s}$ and $t=2.0 \mathrm{s} ?$

Farhanul Hasan

Numerade Educator

Problem 5

(II) A 145 -g baseball, moving along the $x$ axis with speed $30.0 \mathrm{m} / \mathrm{s},$ strikes a fence at a $45^{\circ}$ angle and rebounds along the $y$ axis with unchanged speed. Give its change in momentum using unit vector notation.

Bruce Edelman

Numerade Educator

Problem 6

(II) A 0.145 -kg baseball pitched horizontally at 32.0 $\mathrm{m} / \mathrm{s}$ strikes a bat and is popped straight up to a height of 36.5 $\mathrm{m}$ . If the contact time between bat and ball is 2.5 $\mathrm{ms}$ , calculate the average force between the ball and bat during contact.

Farhanul Hasan

Numerade Educator

Problem 7

(II) A rocket of total mass 3180 $\mathrm{kg}$ is traveling in outer space with a velocity of 115 $\mathrm{m} / \mathrm{s}$ To alter its course by $35.0^{\circ}$ , its rockets can be fired briefly in a direction perpendicular to its original motion. If the rocket gases are expelled at a speed of 1750 $\mathrm{m} / \mathrm{s}$ , how much mass must be expelled?

Bruce Edelman

Numerade Educator

Problem 8

(III) Air in a $120-\mathrm{km} / \mathrm{h}$ wind strikes head-on the face of a building 45 $\mathrm{m}$ wide by 65 $\mathrm{m}$ high and is brought to rest. If air has a mass of 1.3 $\mathrm{kg}$ per cubic meter, determine the average force of the wind on the building.

Farhanul Hasan

Numerade Educator

Problem 9

(I) A 7700 -kg boxcar traveling 18 $\mathrm{m} / \mathrm{s}$ strikes a second car. The two stick together and move off with a speed of 5.0 $\mathrm{m} / \mathrm{s} .$ What is the mass of the second car?

Eric Xue

Numerade Educator

Problem 10

(I) A 9150 -kg railroad car travels alone on a level frictionless track with a constant speed of 15.0 $\mathrm{m} / \mathrm{s}$ . A $4350-\mathrm{kg}$ load, initially at rest, is dropped onto the car. What will be the car's new speed?

Farhanul Hasan

Numerade Educator

Problem 11

(I) An atomic nucleus at rest decays radioactively into an alpha particle and a smaller nucleus. What will be the speed of this recoiling nucleus if the speed of the alpha particle is $2.8 \times 10^{5} \mathrm{m} / \mathrm{s} ?$ Assume the recoiling nucleus has a mass 57 times greater than that of the alpha particle.

Bruce Edelman

Numerade Educator

Problem 12

(I) A 130 -kg tackler moving at 2.5 $\mathrm{m} / \mathrm{s}$ meets head-on (and tackles) an 82 -kg halfback moving at 5.0 $\mathrm{m} / \mathrm{s} .$ What will be their mutual speed immediately after the collision?

Farhanul Hasan

Numerade Educator

Problem 13

(II) A child in a boat throws a 5.70 -kg package out horizontally with a speed of 10.0 $\mathrm{m} / \mathrm{s}$ , Fig. 37 . Calculate the velocity of the boat immediately after, assuming it was initially at rest. The mass of the child is 24.0 $\mathrm{kg}$ and that of the boat is 35.0 $\mathrm{kg}$

Bruce Edelman

Numerade Educator

Problem 14

(II) An atomic nucleus initially moving at 420 $\mathrm{m} / \mathrm{s}$ emits an alpha particle in the direction of its velocity, and the remaining nucleus slows to 350 $\mathrm{m} / \mathrm{s}$ . If the alpha particle has a mass of 4.0 $\mathrm{u}$ and the original nucleus has a mass of $222 \mathrm{u},$ what speed does the alpha particle have when it is emitted?

Farhanul Hasan

Numerade Educator

Problem 15

(II) An object at rest is suddenly broken apart into two fragments by an explosion. One fragment acquires twice the kinetic energy of the other. What is the ratio of their masses?

Bruce Edelman

Numerade Educator

Problem 16

(II) A $22-\mathrm{g}$ bullet traveling 210 $\mathrm{m} / \mathrm{s}$ penetrates a 2.0 -kg block of wood and emerges going 150 $\mathrm{m} / \mathrm{s}$ . If the block is stationary on a frictionless surface when hit, how fast does it move after the bullet emerges?

Farhanul Hasan

Numerade Educator

Problem 17

(II) A rocket of mass $m$ traveling with speed $v_{0}$ along the $x$ axis suddenly shoots out fuel equal to one-third its mass, perpendicular to the $x$ axis (along the $y$ axis) with speed 2$v_{0}$ . Express the final velocity of the rocket in $\hat{\mathbf{j}}, \hat{\mathbf{j}}$ , notation.

Bruce Edelman

Numerade Educator

Problem 18

(II) The decay of a neutron into a proton, an electron, and a neutrino is an example of a three-particle decay process. Use the vector nature of momentum to show that if the neutron is initially at rest, the velocity vectors of the three must be coplanar (that is, all in the same plane). The result is not true for numbers greater than three.

Farhanul Hasan

Numerade Educator

Problem 19

(II) $\mathrm{A}$ mass $m_{\mathrm{A}}=2.0 \mathrm{kg}$ , moving with velocity $\vec{\mathbf{v}}_{\mathrm{A}}=$ $(4.0 \hat{\mathrm{i}}+5.0 \hat{\mathrm{j}}-2.0 \hat{\mathrm{k}}) \mathrm{m} / \mathrm{s},$ collides with mass $m_{\mathrm{B}}=3.0 \mathrm{kg}$ which is initially at rest. Immediately after the collision, mass $m_{\mathrm{A}}$ is observed traveling at velocity $\vec{\mathbf{v}}_{\mathrm{A}}^{\prime}=(-2.0 \hat{\mathrm{i}}+3.0 \hat{\mathbf{k}}) \mathrm{m} / \mathrm{s}$ . Find the velocity of mass $m_{\mathrm{B}}$ after the collision. Assume no outside force acts on the two masses during the collision.

Bruce Edelman

Numerade Educator

Problem 20

(II) $\mathrm{A} 925$ -kg two-stage rocket is traveling at a speed of $6.60 \times 10^{3} \mathrm{m} / \mathrm{s}$ away from Earth when a predesigned explosion separates the rocket into two sections of equal mass that then move with a speed of $2.80 \times 10^{3} \mathrm{m} / \mathrm{s}$ relative to each other along the original line of motion. $(a)$ What is the speed and direction of each section (relative to Earth) after the explosion? $(b)$ How much energy was supplied by the explosion? [Hint. What is the change in kinetic energy as a result of the explosion?]

Farhanul Hasan

Numerade Educator

Problem 21

(III) A 224 -kg projectile, fired with a speed of 116 $\mathrm{m} / \mathrm{s}$ at a $60.0^{\circ}$ angle, breaks into three pieces of equal mass at the highest point of its arc (where its velocity is horizontal). Two of the fragments move with the same speed right after the explosion as the entire projectile had just before the explosion; one of these moves vertically downward and the other horizontally. Determine $(a)$ the velocity of the third fragment immediately after the explosion and $(b)$ the energy released in the explosion.

Bruce Edelman

Numerade Educator

Problem 22

(I) A 0.145 -kg baseball pitched at 35.0 $\mathrm{m} / \mathrm{s}$ is hit on a horizontal line drive straight back at the pitcher at 56.0 $\mathrm{m} / \mathrm{s} .$ If the contact time between bat and ball is $5.00 \times 10^{-3} \mathrm{s}$ , calculate the force (assumed to be constant) between the ball
and bat.

Farhanul Hasan

Numerade Educator

Problem 23

(II) A golf ball of mass 0.045 $\mathrm{kg}$ is hit off the tee at a speed of 45 $\mathrm{m} / \mathrm{s} .$ The golf club was in contact with the ball for $3.5 \times 10^{-3} \mathrm{s} .$ Find $(a)$ the impulse imparted to the golf ball, and $(b)$ the average force exerted on the ball by the golf club.

Bruce Edelman

Numerade Educator

Problem 24

(II) A 12 -kg hammer strikes a nail at a velocity of 8.5 $\mathrm{m} / \mathrm{s}$ and comes to rest in a time interval of 8.0 $\mathrm{ms}(a)$ What is the impulse given to the nail? (b) What is the average force acting on the nail?

Farhanul Hasan

Numerade Educator

Problem 25

(II) A tennis ball of mass $m=0.060 \mathrm{kg}$ and speed $v=25 \mathrm{m} / \mathrm{s}$ strikes a wall at a $45^{\circ}$ angle and rebounds with the same speed at $45^{\circ}$ (Fig. 38). What is the impulse (magnitude and direction) given to the ball?

Bruce Edelman

Numerade Educator

Problem 26

(II) A $130-\mathrm{kg}$ astronaut (including space suit) acquires a speed of 2.50 $\mathrm{m} / \mathrm{s}$ by pushing off with his legs from a 1700 -kg space capsule. (a) What is the change in speed of the space capsule? (b) If the push lasts 0.500 $\mathrm{s}$ , what is the average force exerted by each on the other? As the reference frame, use the position of the capsule before the push. (c) What is the kinetic energy of each after the push?

Farhanul Hasan

Numerade Educator

Problem 27

(II) Rain is falling at the rate of 5.0 $\mathrm{cm} / \mathrm{h}$ and accumulates in a pan. If the raindrops hit at 8.0 $\mathrm{m} / \mathrm{s}$ , estimate the force on the bottom of a 1.0 $\mathrm{m}^{2}$ pan due to the impacting rain which does not rebound. Water has a mass of $1.00 \times 10^{3} \mathrm{kg}$ per $\mathrm{m}^{3}$ .

Bruce Edelman

Numerade Educator

Problem 28

(II) Suppose the force acting on a tennis ball (mass 0.060 $\mathrm{kg}$ ) points in the $+x$ direction and is given by the graph of Fig. 39 as a function of time. Use graphical methods to estimate $(a)$ the total impulse given the ball, and (b) the velocity of the ball after being struck, assuming the ball is being served so it is nearly at rest initially.

Farhanul Hasan

Numerade Educator

Problem 29

(II) With what impulse does a 0.50 -kg newspaper have to be thrown to give it a velocity of 3.0 $\mathrm{m} / \mathrm{s} ?$

Bruce Edelman

Numerade Educator

Problem 30

(II) The force on a bullet is given by the formula $F=\left[740-\left(2.3 \times 10^{5} \mathrm{s}^{-1}\right) t\right] \mathrm{N}$ over the time interval $t=0$ to $t=3.0 \times 10^{-3} \mathrm{s}$ . $(a)$ Plot a graph of $F$ versus $t$ for $t=0$ to $t=3.0 \mathrm{ms}$ . $(b)$ Use the graph to estimate the impulse given the bullet. (c) Determine the impulse by integration. (d) If the bullet achieves a speed of 260 $\mathrm{m} / \mathrm{s}$ as a result of this impulse, given to it in the barrel of a gun, what must the bullet's mass be? (e) What is the recoil speed of the 4.5 -kg gun?

Farhanul Hasan

Numerade Educator

Problem 31

(II) $(a)$ A molecule of mass $m$ and speed $v$ strikes a wall at right angles and rebounds back with the same speed. If the collision time is $\Delta t,$ what is the average force on the wall during the collision? (b) If molecules, all of this type, strike the wall at intervals a time $t$ apart (on the average) what is the average force on the wall averaged over a long time?

Bruce Edelman

Numerade Educator

Problem 32

(III) (a) Calculate the impulse experienced when a $65-\mathrm{kg}$ person lands on firm ground after jumping from a height of 3.0 $\mathrm{m}$ . (b) Estimate the average force exerted on the person's feet by the ground if the landing is stiff-legged, and again (c) with bent legs. With stiff legs, assume the body moves 1.0 $\mathrm{cm}$ during impact, and when the legs are bent, about 50 $\mathrm{cm} .$ [Hint. The average net force on her which is related to impulse, is the vector sum of gravity and the force exerted by the ground.]

Farhanul Hasan

Numerade Educator

Problem 33

(III) A scale is adjusted so that when a large, shallow pan is placed on it, it reads zero. A water faucet at height $h=2.5 \mathrm{m}$ above is turned on and water falls into the pan at a rate $R=0.14 \mathrm{kg} / \mathrm{s}$ . Determine $(a)$ a formula for the scale reading as a function of time $t$ and $(b)$ the reading for $t=9.0 \mathrm{s}(c)$ Repeat $(a)$ and $(b),$ but replace the shallow pan with a tall, narrow cylindrical container of area $A=20 \mathrm{cm}^{2}$ (the level rises in this case).

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Problem 34

(II) A $0.060-\mathrm{kg}$ tennis ball, moving with a speed of 4.50 $\mathrm{m} / \mathrm{s}$ , has a head-on collision with a $0.090-\mathrm{kg}$ ball initially moving in the same direction at a speed of 3.00 $\mathrm{m} / \mathrm{s}$ . Assuming a perfectly elastic collision, determine the speed and direction of each ball after the collision.

Farhanul Hasan

Numerade Educator

Problem 35

(II) A $0.450-\mathrm{kg}$ hockey puck, moving east with a speed of $4.80 \mathrm{m} / \mathrm{s},$ has a head-on collision with a $0.900-\mathrm{kg}$ puck initially at rest. Assuming a perfectly elastic collision, what will be the speed and direction of each object after the collision?

Bruce Edelman

Numerade Educator

Problem 36

(II) A $0.280-\mathrm{kg}$ croquet ball makes an elastic head-on collision with a second ball initially at rest. The second ball moves off with half the original speed of the first ball. (a) What is the mass of the second ball? (b) What fraction of the original kinetic energy $(\Delta K / K)$ gets transferred to the second ball?

Farhanul Hasan

Numerade Educator

Problem 37

(II) A ball of mass 0.220 $\mathrm{kg}$ that is moving with a speed of 7.5 $\mathrm{m} / \mathrm{s}$ collides head-on and elastically with another ball initially at rest. Immediately after the collision, the incoming ball bounces backward with a speed of 3.8 $\mathrm{m} / \mathrm{s}$ . Calculate (a) the velocity of the target ball after the collision, and (b) the mass of the target ball.

Bruce Edelman

Numerade Educator

Problem 38

(II) A ball of mass $m$ makes a head-on elastic collision with a second ball (at rest) and rebounds with a speed equal to 0.350 its original speed. What is the mass of the second ball?

Farhanul Hasan

Numerade Educator

Problem 39

(II) Determine the fraction of kinetic energy lost by a neutron $\left(m_{1}=1.01 \mathrm{u}\right)$ when it collides head-on and elastically with a target particle at rest which is $(a)$ i $\mathbf{H}(m=1.01 \mathrm{u}) ;$ (b) $^{2} \mathrm{H}$ (heavy hydrogen, $m=2.01 \mathrm{u} ) ;(c) \frac{12}{6} \mathrm{C}(m=12.00 \mathrm{u}) ;$ (d) $\frac{208 \mathrm{Pb}}{82 \mathrm{Pb}}($ lead $, m=208 \mathrm{u})$

Bruce Edelman

Numerade Educator

Problem 40

(II) Show that, in general, for any head-on one-dimensional elastic collision, the speeds after collision are
$v_{\mathrm{B}}^{\prime}=v_{\mathrm{A}}\left(\frac{2 m_{\mathrm{A}}}{m_{\mathrm{A}}+m_{\mathrm{B}}}\right)+v_{\mathrm{B}}\left(\frac{m_{\mathrm{B}}-m_{\mathrm{A}}}{m_{\mathrm{A}}+m_{\mathrm{B}}}\right)$
and
$v_{\mathrm{A}}^{\prime}=v_{\mathrm{A}}\left(\frac{m_{\mathrm{A}}-m_{\mathrm{B}}}{m_{\mathrm{A}}+m_{\mathrm{B}}}\right)+v_{\mathrm{B}}\left(\frac{2 m_{\mathrm{B}}}{m_{\mathrm{A}}+m_{\mathrm{B}}}\right)$
where $v_{\mathrm{A}}$ and $v_{\mathrm{B}}$ are the initial speeds of the two objects of mass $m_{\mathrm{A}}$ and $m_{\mathrm{B}} .$

Farhanul Hasan

Numerade Educator

Problem 41

(III) $\mathrm{A} 3.0$ -kg block slides along a frictionless tabletop at 8.0 $\mathrm{m} / \mathrm{s}$ toward a second block (at rest) of mass 4.5 $\mathrm{kg}$ . A coil spring, which obeys Hooke's law and has spring constant $k=850 \mathrm{N} / \mathrm{m},$ is attached to the second block in such a way that it will be compressed when struck by the moving block, Fig. 40 (a) What will be the maximum compression of the spring? (b) What will be the final velocities of the blocks after the collision? (c) Is the collision elastic? Ignore the mass of the spring.

Bruce Edelman

Numerade Educator

Problem 42

(I) In a ballistic pendulum experiment, projectile 1 results in a maximum height $h$ of the pendulum equal to 2.6 $\mathrm{cm} . \mathrm{A}$ second projectile (of the same mass) causes the pendulum to swing twice as high, $h_{2}=5.2 \mathrm{cm} .$ The second projectile was how many times faster than the first?

Farhanul Hasan

Numerade Educator

Problem 43

(II) $(a)$ Derive a formula for the fraction of kinetic energy lost, $\Delta K / K,$ in terms of $m$ and $M$ for the ballistic pendulum collision of Example 11 of "Linear Momentum". (b) Evaluate for $m=16.0 \mathrm{g}$ and $M=380 \mathrm{g}$ .

Bruce Edelman

Numerade Educator

Problem 44

(II) $\mathrm{A} 28$ -g rifle bullet traveling 210 $\mathrm{m} / \mathrm{s}$ buries itself in a 3.6 $\mathrm{-kg}$ pendulum hanging on a 2.8 -long string, which makes the pendulum swing upward in an arc. Determine the vertical and horizontal components of the pendulum's maximum displacement.

Farhanul Hasan

Numerade Educator

Problem 45

(II) An internal explosion breaks an object, initially at rest, into two pieces, one of which has 1.5 times the mass of the other. If 7500 $\mathrm{J}$ is released in the explosion, how much kinetic energy does each piece acquire?

WP

Wayne Peterson

Numerade Educator

Problem 46

(II) $\mathrm{A} 920$ -kg sports car collides into the rear end of a 2300 -kg SUV stopped at a red light. The bumpers lock, the brakes are locked, and the two cars skid forward 2.8 $\mathrm{m}$ before stopping. The police officer, estimating the coefficient of kinetic friction between tires and road to be 0.80 , calculates the speed of the sports car at impact. What was that speed?

Farhanul Hasan

Numerade Educator

Problem 47

(II) You drop a 12 -g ball from a height of 1.5 $\mathrm{m}$ and it only bounces back to a height of 0.75 $\mathrm{m} .$ What was the total impulse on the ball when it hit the floor? (Ignore air resistance).

Bruce Edelman

Numerade Educator

Problem 48

(II) Car A hits car $\mathrm{B}$ (initially at rest and of equal mass) from behind while going 35 $\mathrm{m} / \mathrm{s} .$ Immediately after the collision, car $\mathrm{B}$ moves forward at 25 $\mathrm{m} / \mathrm{s}$ and car $\mathrm{A}$ is at rest. What fraction of the initial kinetic energy is lost in the collision?

Farhanul Hasan

Numerade Educator

Problem 49

(II) A measure of inelasticity in a head-on collision of two objects is the coefficient of restitution, $e$ , defined as
$e=\frac{v_{\mathrm{A}}^{\prime}-v_{\mathrm{B}}^{\prime}}{v_{\mathrm{B}}-v_{\mathrm{A}}}$
where $v_{\mathrm{A}}^{\prime}-v_{\mathrm{B}}^{\prime}$ is the relative velocity of the two objects after the collision and $v_{\mathrm{B}}-v_{\mathrm{A}}$ is their relative velocity before it. $(a)$ Show that $e=1$ for a perfectly elastic collision, and $e=0$ for a completely inelastic collision. $(b)$ A simple method for measuring the coefficient of restitution for an object colliding with a very hard surface like steel is to drop the object onto a heavy steel plate, as shown in Fig. $41 .$ Determine a formula for $e$ in terms of the original height $h$ and the maximum height $h^{\prime}$ reached after collision.

Bruce Edelman

Numerade Educator

Problem 50

(II) A pendulum consists of a mass $M$ hanging at the bottom end of a massless rod of length $\ell,$ which has a frictionless pivot at its top end. A mass $m,$ moving as shown in Fig. 42 with velocity $v$
impacts $M$ and becomes embedded. What is the smallest value of $v$ sufficient to cause the pendulum (with embedded mass $m$ ) to swing clear over the top of its arc?

Farhanul Hasan

Numerade Educator

Problem 51

(II) A bullet of mass $m=0.0010 \mathrm{kg}$ embeds itself in a wooden block with mass $M=0.999 \mathrm{kg},$ which then compresses a spring $(k=120 \mathrm{N} / \mathrm{m})$ by a distance $x=0.050 \mathrm{m}$ before coming to rest. The coefficient of kinetic friction between the block and table is $\mu=0.50 .$ (a) What is the initial speed of the bullet? (b) What fraction of the bullet's initial kinetic energy is dissipated (in damage to the wooden block, rising temperature, etc. $.$ in the collision between the bullet and the block?

Bruce Edelman

Numerade Educator

Problem 52

(II) A 144 -g baseball moving 28.0 $\mathrm{m} / \mathrm{s}$ strikes a stationary 5.25 -kg brick resting on small rollers so it moves without significant friction. After hitting the brick, the baseball bounces straight back, and the brick moves forward at 1.10 $\mathrm{m} / \mathrm{s} .(a)$ What is the baseball's speed after the collision? (b) Find the total kinetic energy before and after the collision.

Farhanul Hasan

Numerade Educator

Problem 53

(II) A 6.0 -kg object moving in the $+x$ direction at 5.5 $\mathrm{m} / \mathrm{s}$ collides head-on with an $8.0-\mathrm{kg}$ object moving in the $-x$ direction at 4.0 $\mathrm{m} / \mathrm{s} .$ Find the final velocity of each mass if: $(a)$ the objects stick together; $(b)$ the collision is elastic; $(c)$ the $6.0-\mathrm{kg}$ object is at rest after the collision; $(d)$ the 8.0 -kg object is at rest after the collision; $(e)$ the 6.0 -kg object has a velocity of 4.0 $\mathrm{m} / \mathrm{s}$ in the $-x$ direction after the collision. Are the results in $(c),(d),$ and $(e)$ "reasonable"? Explain.

Ashly Sunny

Numerade Educator

Problem 54

(II) Billiard ball A of mass $m_{\mathrm{A}}=0.120 \mathrm{kg}$ moving with speed $v_{\mathrm{A}}=2.80 \mathrm{m} / \mathrm{s}$ strikes ball $\mathrm{B}$ , initially at rest, of mass $m_{\mathrm{B}}=0.140 \mathrm{kg} .$ As a result of the collision, ball $\mathrm{A}$ is deflected off at an angle of $30.0^{\circ}$ with a speed $v_{\mathrm{A}}^{\prime}=2.10 \mathrm{m} / \mathrm{s}$ (a) Taking the $x$ axis to be the original direction of motion of ball $A,$ write down the equations expressing the conservation of momentum for the components in the $x$ and $y$ directions separately. (b) Solve these equations for the speed, $v_{\mathrm{B}}^{\prime},$ and angle, $\theta_{\mathrm{B}}^{\prime},$ of ball B. Do not assume the collision is elastic.

Farhanul Hasan

Numerade Educator

Problem 55

(II) A radioactive nucleus at rest decays into a second nucleus, an electron, and a neutrino. The electron and neutrino are emitted at right angles and have momenta of $9.6 \times 10^{-23} \mathrm{kg} \cdot \mathrm{m} / \mathrm{s}$ and $6.2 \times 10^{-23} \mathrm{kg} \cdot \mathrm{m} / \mathrm{s},$ respectively. Determine the magnitude and the direction of the momentum of the second (recoiling) nucleus.

Bruce Edelman

Numerade Educator

Problem 56

(II) Two billiard balls of equal mass move at right angles and meet at the origin of an $x y$ coordinate system. Initially ball $A$ is moving along the $y$ axis at $+2.0 \mathrm{m} / \mathrm{s},$ and ball $\mathrm{B}$ is moving to the right along the $x$ axis with speed $+3.7 \mathrm{m} / \mathrm{s}$ . After the collision (assumed elastic), the second ball is moving along the positive $y$ axis (Fig. $43 ) .$ What is the final direction of ball $A,$ and what are the speeds of the two balls?

Farhanul Hasan

Numerade Educator

Problem 57

(II) An atomic nucleus of mass $m$ traveling with speed $v$ collides elastically with a target particle of mass 2$m$ (initially at rest) and is scattered at $90^{\circ} .(a)$ At what angle does the target particle move after the collision? (b) What are the final speeds of the two particles? (c) What fraction of the initial kinetic energy is transferred to the target particle?

Bruce Edelman

Numerade Educator

Problem 58

(II) A neutron collides elastically with a helium nucleus (at rest initially) whose mass is four times that of the neutron. The helium nucleus is observed to move off at an angle $\theta_{\mathrm{Hc}}^{\prime}=45^{\circ} .$ Determine the angle of the neutron, $\theta_{\mathrm{n}}^{\prime},$ and the speeds of the two particles, $v_{\mathrm{n}}^{\prime}$ and $v_{\mathrm{He}}^{\prime}$ , after the collision. The neutron's initial speed is $6.2 \times 10^{5} \mathrm{m} / \mathrm{s}$ .

Farhanul Hasan

Numerade Educator

Problem 59

(III) A neon atom $(m=20.0 \mathrm{u})$ makes a perfectly elastic collision with another atom at rest. After the impact, the neon atom travels away at a $55.6^{\circ}$ angle from its original direction and the unknown atom travels away at a $-50.0^{\circ}$ angle. What is the mass $($ in u ) of the unknown atom?
[Hint: You could use the law of sines.]

Bruce Edelman

Numerade Educator

Problem 60

(III) For an elastic collision between a projectile particle of mass $m_{\mathrm{A}}$ and a target particle (at rest) of mass $m_{\mathrm{B}},$ show that the scattering angle, $\theta_{\mathrm{A}}^{\prime},$ of the projectile $(a)$ can take any value, 0 to $180^{\circ},$ for $m_{A}<m_{B},$ but $(b)$ has a maximum
angle $\phi$ given by $\cos ^{2} \phi=1-\left(m_{B} / m_{A}\right)^{2}$ for $m_{A}>m_{B}$ .

Farhanul Hasan

Numerade Educator

Problem 61

(III) Prove that in the elastic collision of two objects of identical mass, with one being a target initially at rest, the angle between their final velocity vectors is always $90^{\circ} .$

Bruce Edelman

Numerade Educator

Problem 62

(1) The CM of an empty $1250-\mathrm{kg}$ car is 2.50 $\mathrm{m}$ behind the front of the car. How far from the front of the car will the cM be when two people sit in the front seat 2.80 $\mathrm{m}$ from the front of the car, and three people sit in the back seat 3.90 $\mathrm{m}$ from the front? Assume that each person has a mass of 70.0 $\mathrm{kg}$ .

Farhanul Hasan

Numerade Educator

Problem 63

(1) The distance between a carbon atom $(m=12 \mathrm{u})$ and an oxygen atom $(m=16 \mathrm{u})$ in the $\mathrm{CO}$ molecule is $1.13 \times 10^{-10} \mathrm{m} .$ How far from the carbon atom is the center of mass of the molecule?

Bruce Edelman

Numerade Educator

Problem 64

(II) Three cubes, of side $\ell_{0}, 2 \ell_{0},$ and $3 \ell_{0},$ are placed next to one another (in contact) with their centers along a straight line as shown in Fig. $44 .$ What is the position, along this line, of the CM of this system? Assume the cubes are made of the same uniform material.

Farhanul Hasan

Numerade Educator

Problem 65

A particle of mass $1.00 \mathrm{~kg}$ is moving with velocity $\vec{v}=(7.0 \mathrm{i}+6.0 \mathrm{j}) \mathrm{m} / \mathrm{s} . \quad(a)$ Find the angular momentum
1. relative to the origin when the particle is at $\overrightarrow{\mathbf{r}}=(2.0 \mathbf{j}+4.0 \mathbf{k}) \mathrm{m}$
(b) At position $\overrightarrow{\mathbf{r}}$ a force of $\mathbf{F}=4.0 \mathrm{Ni}$ is applied to the particle. Find the torque relative to the origin.

Bruce Edelman

Numerade Educator

Problem 66

(II) A uniform circular plate of radius 2$R$ has a circular hole of radius $R$ cut out of it. The center $C^{\prime}$ of the smaller circle is a distance 0.80$R$ from the center $C$ of the larger circle, Fig. $45 .$ What is the position of the center of mass of the plate? [Hint: Try subtraction.]

Farhanul Hasan

Numerade Educator

Problem 67

(II) A uniform thin wire is bent into a semicircle of radius $r .$ Determine the coordinates of its center of mass with respect to an origin of coordinates at the center of the "full" circle.

Bruce Edelman

Numerade Educator

Problem 68

(II) Find the center of mass of the ammonia molecule. The chemical formula is $\mathrm{NH}_{3}$ . The hydrogens are at the corners of an equilateral triangle (with sides 0.16 $\mathrm{nm} )$ that forms the base of a pyramid, with nitrogen at the apex $(0.037 \mathrm{nm}$ vertically above the plane of the triangle).

Farhanul Hasan

Numerade Educator

Problem 69

(III) Determine the $\mathrm{CM}$ of a machine part that is a uniform cone of height $h$ and radius $R,$ Fig. $46 .$ [Hint: Divide the cone into an infinite number of disks of thickness $d z,$ one of which is shown.

Bruce Edelman

Numerade Educator

Problem 70

(III) Determine the $\mathrm{CM}$ of a uniform pyramid that has four triangular faces and a square base with equal sides all of length $s .$ [Hint: See Problem 69.]

Farhanul Hasan

Numerade Educator

Problem 71

(III) Determine the $\mathrm{CM}$ of a thin, uniform, semicircular plate.

Bruce Edelman

Numerade Educator

Problem 72

(II) Mass $M_{A}=35 \mathrm{kg}$ and mass $M_{\mathrm{B}}=25 \mathrm{kg} .$ They have velocities $($ in $\mathrm{m} / \mathrm{s}) \vec{\mathbf{v}}_{\mathrm{A}}=12 \hat{\mathbf{i}}-16 \hat{\mathrm{j}}$ and $\vec{\mathbf{v}}_{\mathrm{B}}=-20 \hat{\mathbf{i}}+14 \hat{\mathbf{j}}$ Determine the velocity of the center of mass of the system.

Farhanul Hasan

Numerade Educator

Problem 73

(II) The masses of the Earth and Moon are $5.98 \times 10^{24} \mathrm{kg}$ and $7.35 \times 10^{22} \mathrm{kg},$ respectively, and their centers are separated by $3.84 \times 10^{8} \mathrm{m}$ . (a) Where is the CM of this system located? (b) What can you say about the motion of the Earth-Moon system about the Sun, and of the Earth and Moon separately about the Sun?

Bruce Edelman

Numerade Educator

Problem 74

(II) A mallet consists of a uniform cylindrical head of mass 2.80 $\mathrm{kg}$ and a diameter 0.0800 $\mathrm{m}$ mounted on a uniform cylindrical handle of mass 0.500 $\mathrm{kg}$ and length $0.240 \mathrm{m},$ as shown in Fig. $47 .$ If this mallet is tossed, spinning, into the air, how far above the bottom of the handle is the point that will follow a parabolic trajectory?

Farhanul Hasan

Numerade Educator

Problem 75

(II) A 55 -kg woman and a $72-\mathrm{kg}$ man stand 10.0 $\mathrm{m}$ apart on frictionless ice. (a) How far from the woman is their $\mathrm{cm}$ ? (b) If each holds one end of a rope, and the man pulls on the rope so that he moves $2.5 \mathrm{m},$ how far from the woman will he be now? (c) How far will the man have moved when he collides with the woman?

Bruce Edelman

Numerade Educator

Problem 76

(II) Suppose that in Example 18 of "Linear Momentum" (Fig. $32 ), m_{11}=3 m_{1} .(a)$ Where then would $m_{11}$ land? $(b)$ What if $m_{1}=3 m_{11} ?$

Farhanul Hasan

Numerade Educator

Problem 77

(II) Two people, one of mass 85 $\mathrm{kg}$ and the other of mass $55 \mathrm{kg},$ sit in a rowboat of mass 78 $\mathrm{kg}$ . With the boat initially at rest, the two people, who have been sitting at opposite ends of the boat, 3.0 $\mathrm{m}$ apart from each other, now exchange seats. How far and in what direction will the boat move?

Bruce Edelman

Numerade Educator

Problem 78

(III) A $280-\mathrm{kg}$ flatcar 25 $\mathrm{m}$ long is moving with a speed of 6.0 $\mathrm{m} / \mathrm{s}$ along horizontal frictionless rails. A $95-\mathrm{kg}$ worker starts walking from one end of the car to the other in the direction of motion, with speed 2.0 $\mathrm{m} / \mathrm{s}$ with respect to the car. In the time it takes for him to reach the other end, how far has the flatcar moved?

Farhanul Hasan

Numerade Educator

Problem 79

(III) A huge balloon and its gondola, of mass $M,$ are in the air and stationary with respect to the ground. A passenger, of mass $m,$ then climbs out and slides down a rope with speed $v,$ measured with respect to the balloon. With what speed and direction (relative to Earth) does the balloon then move? What happens if the passenger stops?

Bruce Edelman

Numerade Educator

Problem 80

(II) $\mathrm{A} 3500$ -kg rocket is to be accelerated at 3.0 $\mathrm{g}$ at take-off from the Earth. If the gases can be ejected at a rate of $27 \mathrm{kg} / \mathrm{s},$ what must be their exhaust speed?

Farhanul Hasan

Numerade Educator

Problem 81

(II) Suppose the conveyor belt of Example 19 of "Linear Momentum" is retarded by a friction force of 150 $\mathrm{N}$ . Determine the required output power (hp) of the motor as a function of time from the moment gravel first starts falling $(t=0)$ until 3.0 s after the gravel begins to be dumped off the end of the 22 -m-long conveyor belt.

Bruce Edelman

Numerade Educator

Problem 82

(II) The jet engine of an airplane takes in 120 $\mathrm{kg}$ of air per second, which is burned with 4.2 $\mathrm{kg}$ of fuel per second.The burned gases leave the plane at a speed of 550 $\mathrm{m} / \mathrm{s}$ (relative to the plane). If the plane is traveling $270 \mathrm{m} / \mathrm{s}(600 \mathrm{mi} / \mathrm{h}),$ determine: $(a)$ the thrust due to ejected fuel; $(b)$ the thrust due to accelerated air passing through the engine; and $(c)$ the power (hp) delivered.

Farhanul Hasan

Numerade Educator

Problem 83

(II) A rocket traveling 1850 $\mathrm{m} / \mathrm{s}$ away from the Earth at an altitude of 6400 $\mathrm{km}$ fires its rockets, which eject gas at a speed of 1300 $\mathrm{m} / \mathrm{s}$ (relative to the rocket). If the mass of the rocket at this moment is $25,000 \mathrm{kg}$ and an acceleration of 1.5 $\mathrm{m} / \mathrm{s}^{2}$ is desired, at what rate must the gases be ejected?

Bruce Edelman

Numerade Educator

Problem 84

(III) A sled filled with sand slides without friction down a $32^{\circ}$ slope. Sand leaks out a hole in the sled at a rate of 2.0 $\mathrm{kg} / \mathrm{s}$ . If the sled starts from rest with an initial total mass of 40.0 $\mathrm{kg}$ . how long does it take the sled to travel 120 $\mathrm{m}$ along the slope?

Farhanul Hasan

Numerade Educator

Problem 85

A novice pool player is faced with the corner pocket shot shown in Fig. $48 .$ Relative dimensions are also shown. Should the player worry that this might be a "scratch shot," in which the cue ball will also fall into a pocket? Give details. Assume equal mass balls and an elastic collision.

Bruce Edelman

Numerade Educator

Problem 86

During a Chicago storm, winds can whip horizontally at speeds of 120 $\mathrm{km} / \mathrm{h}$ . If the air strikes a person at the rate of 45 $\mathrm{kg} / \mathrm{s}$ per square meter and is brought to rest, calculate the force of the wind on a person. Assume the person is 1.60 $\mathrm{m}$ high and 0.50 $\mathrm{m}$ wide. Compare to the typical maximum force of friction $(\mu \approx 1.0)$ between the person and the ground, if the person has a mass of 75 $\mathrm{kg}$ .

Farhanul Hasan

Numerade Educator

Problem 87

A ball is dropped from a height of 1.50 $\mathrm{m}$ and rebounds to a height of 1.20 $\mathrm{m} .$ Approximately how many rebounds will the ball make before losing 90$\%$ of its energy?

Bruce Edelman

Numerade Educator

Problem 88

In order to convert a tough split in bowling, it is necessary to strike the pin a glancing blow as shown in Fig. $49.49 .$ Assume that the bowling ball, initially traveling at $13.0 \mathrm{m} / \mathrm{s},$ has five times the mass of a pin and that the pin goes off at $75^{\circ}$ from the original direction of the ball. Calculate the speed (a) of the pin and $(b)$ of the ball just after collision, and $(c)$ calculate the angle through which the ball was deflected. Assume the collision is elastic and ignore any spin of the ball.

Farhanul Hasan

Numerade Educator

Problem 89

A gun fires a bullet vertically into a $1.40-$ kg block of wood at rest on a thin horizontal sheet, Fig. $50 .$ If the bullet has a mass of 24.0 $\mathrm{g}$ and a speed of $310 \mathrm{m} / \mathrm{s},$ how high will the block rise into the air after the bullet becomes embedded in it?

Bruce Edelman

Numerade Educator

Problem 90

A hockey puck of mass 4 $\mathrm{m}$ has been rigged to explode, as part of a practical joke. Initially the puck is at rest on a frictionless ice rink. Then it bursts into three pieces. One chunk, of mass $m,$ slides across the ice at velocity $v \hat{\mathbf{i}}$ . Another chunk, of mass $2 m,$ slides across the ice at velocity 2$v \hat{\mathbf{j}}$ . Determine the velocity of the third chunk.

Farhanul Hasan

Numerade Educator

Problem 91

For the completely inelastic collision of two railroad cars that we considered in Example 3 of "Linear Momentum," calculate how much of the initial kinetic energy is transformed to thermal or other forms of energy.

Bruce Edelman

Numerade Educator

Problem 92

A 4800 -kg open railroad car coasts along with a constant speed of 8.60 $\mathrm{m} / \mathrm{s}$ on a level track. Snow begins to fall vertically and fills the car at a rate of 3.80 $\mathrm{kg} / \mathrm{min}$ . Ignoring friction with the tracks, what is the speed of the car after 60.0 $\mathrm{min}$ ? (See Section 2 of "Linear Momentum.")

Farhanul Hasan

Numerade Educator

Problem 93

Consider the railroad car of Problem $92,$ which is slowly filling with snow. (a) Determine the speed of the car as a function of time using Eqs. $19 .$ (b) What is the speed of the car after 60.0 min? Does this agree with the simpler calculation (Problem 92$) ?$
$M \frac{d \vec{\mathbf{v}}}{d t}=\Sigma \vec{\mathbf{F}}_{\mathrm{ext}}+\vec{\mathbf{v}}_{\mathrm{rel}} \frac{d M}{d t}$

Bruce Edelman

Numerade Educator

Problem 94

Two blocks of mass $m_{\mathrm{A}}$ and $m_{\mathrm{B}},$ resting on a frictionless table, are connected by a stretched spring and then released (Fig. $51 ) .(a)$ Is there a net external force on the system? (b) Determine the ratio of their speeds, $v_{A} / v_{B}$ . (c) What is the ratio of their kinetic energies? (d) Describe the motion of the $\mathrm{CM}$ of this system. (e) How would the presence of friction alter the above results?

Farhanul Hasan

Numerade Educator

Problem 95

You have been hired as an expert witness in a court case involving an automobile accident. The accident involved car A of mass 1500 $\mathrm{kg}$ which crashed into stationary car $\mathrm{B}$ of mass 1100 $\mathrm{kg} .$ The driver of car A applied his brakes 15 $\mathrm{m}$ before he skidded and crashed into car B. After the collision, car $A$ slid 18 $\mathrm{m}$ while car $\mathrm{B}$ slid 30 $\mathrm{m}$ . The coefficient of kinetic friction between the locked wheels and the road was measured to be $0.60 .$ Show that the driver of car A was exceeding the $55-\mathrm{mi} / \mathrm{h}(90 \mathrm{km} / \mathrm{h})$ speed limit before applying the brakes.

Bruce Edelman

Numerade Educator

Problem 96

A meteor whose mass was about $2.0 \times 10^{8} \mathrm{kg}$ struck the Earth $\left(m_{\mathrm{E}}=6.0 \times 10^{24} \mathrm{kg}\right)$ with a speed of about 25 $\mathrm{km} / \mathrm{s}$ and came to rest in the Earth. (a) What was the Earth's recoil speed (relative to Earth at rest before the collision)? b) What fraction of the meteor's kinetic energy was transformed to kinetic energy of the Earth? (c) By how much did the Earth's kinetic energy change as a result of this collision?

Farhanul Hasan

Numerade Educator

Problem 97

Two astronauts, one of mass 65 $\mathrm{kg}$ and the other $85 \mathrm{kg},$ are initially at rest in outer space. Then then push each other apart. How far apart are they when the lighter astronaut has moved 12 $\mathrm{m} ?$

Bruce Edelman

Numerade Educator

Problem 98

A 22 -g bullet strikes and becomes embedded in a 1.35 -kg block of wood placed on a horizontal surface just in front of the gun. If the coefficient of kinetic friction between the block and the surface is $0.28,$ and the impact drives the block a distance of 8.5 $\mathrm{m}$ before it comes to rest, what was the muzzle speed of the bullet?

Farhanul Hasan

Numerade Educator

Problem 99

Two balls, of masses $m_{\mathrm{A}}=45 \mathrm{g}$ and $m_{\mathrm{B}}=65 \mathrm{g},$ are suspended as shown in Fig. $52 .$ The lighter ball is pulled away to a $66^{\circ}$ angle with the vertical and released.
(a) What is the velocity of the lighter ball before impact?
(b) What is the velocity of each ball after the elastic collision?
(c) What will be the maximum height of each ball after the elastic collision?

Kristela Garcia

Kristela Garcia

Numerade Educator

Problem 100

A block of mass $m=2.20 \mathrm{kg}$ slides down a $30.0^{\circ}$ incline which is 3.60 $\mathrm{m}$ high. At the bottom, it strikes a block of mass $M=7.00 \mathrm{kg}$ which is at rest on a horizontal surface, Fig. 53 . (Assume a smooth transition at the bottom of the incline.) If the collision is elastic, and friction can be ignored, determine $(a)$ the speeds of the two blocks after the collision, and (b) how far back up the incline the smaller mass will go.

Farhanul Hasan

Numerade Educator

Problem 101

In Problem 100 (Fig. 53), what is the upper limit on mass $m$ if it is to rebound from $M,$ slide up the incline, stop, slide down the incline, and collide with $M$ again?

Bruce Edelman

Numerade Educator

Problem 102

After a completely inelastic collision between two objects of equal mass, each having initial speed, $v,$ the two move off together with speed $v / 3 .$ What was the angle between their initial directions?

Farhanul Hasan

Numerade Educator

Problem 103

A 0.25 -kg skeet (clay target) is fired at an angle of $28^{\circ}$ to the horizon with a speed of 25 $\mathrm{m} / \mathrm{s}$ (Fig. $54 ) .$ When it reaches the maximum height, $h$ , it is hit from below by a $15-\mathrm{g}$ pellet traveling vertically upward at a speed of 230 $\mathrm{m} / \mathrm{s}$ .
The pellet is embedded in the skeet. (a) How much higher, $h^{\prime},$ did the skeet go up? (b) How much extra distance, $\Delta x$ does the skeet travel because of the collision?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 104

A massless spring with spring constant $k$ is placed between a block of mass $m$ and a block of mass 3$m .$ Initially the blocks are at rest on a frictionless surface and they are held together so that the spring between them is compressed by an amount $D$ from its equilibrium length. The blocks are then released and the spring pushes them off in opposite directions. Find the speeds of the two blocks when they detach from the spring.

Farhanul Hasan

Numerade Educator

Problem 105

The gravitational slingshot effect. Figure 55 shows the planet Saturn moving in the negative $x$ direction at its orbital speed (with respect to the Sun) of 9.6 $\mathrm{km} / \mathrm{s}$ . The mass of Saturn is $5.69 \times 10^{26} \mathrm{kg} .$ A spacecraft with mass 825 $\mathrm{kg}$ approaches Saturn. When far from Saturn, it moves in the $+x$ direction at 10.4 $\mathrm{km} / \mathrm{s} .$ The gravitational attraction of Saturn (a conservative force) acting on the spacecraft causes it to swing around the planet (orbit shown as dashed line ) and head off in the opposite direction. Estimate the final speed of the spacecraft after it is far enough away to be considered free of Saturn's gravitational pull.

Bruce Edelman

Numerade Educator

Problem 106

Two bumper cars in an amusement park ride collide elastically as one approaches the other directly from the rear (Fig. 56). Car A has a mass of 450 $\mathrm{kg}$ and car $\mathrm{B} 490 \mathrm{kg},$ owing to differences in passenger mass. If car A approaches at 4.50 $\mathrm{m} / \mathrm{s}$ and car $\mathrm{B}$ is moving at 3.70 $\mathrm{m} / \mathrm{s}$ , calculate $(a)$ their velocities after the collision, and (b) the change in momentum of each.

Farhanul Hasan

Numerade Educator

Problem 107

In a physics lab, a cube slides down a frictionless incline as shown in Fig. 57 and elastically strikes another cube at the bottom that is only one-half its mass. If the incline is 35 $\mathrm{cm}$ high and
the table is 95 $\mathrm{cm}$ off the floor, where does each cube land? [Hint. Both leave the incline moving horizontally.]

Bruce Edelman

Numerade Educator

Problem 108

The space shuttle launches an $850-$ kg satellite by ejecting it from the cargo bay. The ejection mechanism is activated and is in contact with the satellite for 4.0 s to give it a velocity of 0.30 $\mathrm{m} / \mathrm{s}$ in the $z$ -direction relative to the shuttle. The mass of the shuttle is $92,000 \mathrm{kg}$ . (a) Determine the component of velocity $v_{\mathrm{f}}$ of the shuttle in the minus z-direction resulting from the ejection. $(b)$ Find the average force that the shuttle exerts on the satellite during the ejection.

Farhanul Hasan

Numerade Educator

Problem 109

You are the design engineer in charge of the crashworthiness of new automobile models. Cars are tested by smashing them into fixed, massive barriers at 45 $\mathrm{km} / \mathrm{h} .$ A new model of mass 1500 $\mathrm{kg}$ takes 0.15 s from the time of impact until it is brought to rest. (a) Calculate the average force exerted on the car by the barrier. (b) Calculate the average deceleration of the car.

Bruce Edelman

Numerade Educator

Problem 110

Astronomers estimate that a $2.0-\mathrm{km}$ -wide asteroid collides with the Earth once every million years. The collision could pose a threat to life on Earth. (a) Assume a spherical asteroid has a mass of 3200 $\mathrm{kg}$ for each cubic meter of volume and moves toward the Earth at 15 $\mathrm{km} / \mathrm{s} .$ How much destructive energy could be released when it embeds itself in the Earth? (b) For comparison, a nuclear bomb could release about $4.0 \times 10^{16} \mathrm{J} .$ How many such bombs would have to explode simultaneously to release the destructive energy of the asteroid collision with the Earth?

Farhanul Hasan

Numerade Educator

Problem 111

An astronaut of mass 210 $\mathrm{kg}$ including his suit and jet pack wants to acquire a velocity of 2.0 $\mathrm{m} / \mathrm{s}$ to move back toward his space shuttle. Assuming the jet pack can eject gas with a velocity of $35 \mathrm{m} / \mathrm{s},$ what mass of gas will need to be ejected?

Bruce Edelman

Numerade Educator

Problem 112

An extrasolar planet can be detected by observing the wobble it produces on the star around which it revolves Suppose an extrasolar planet of mass $m_{B}$ revolves around its star of mass $m_{A}$ . If no external force acts on this simple two-object system, then its $\mathrm{CM}$ is stationary. Assume $m_{\mathrm{A}}$ and $m_{\mathrm{B}}$ are in circular orbits with radii $r_{\mathrm{A}}$ and $r_{\mathrm{B}}$ about the system's $\mathrm{CM} .(a)$ Show that
$r_{\mathrm{A}}=\frac{m_{\mathrm{B}}}{m_{\mathrm{A}}} r_{\mathrm{B}}$
(b) Now consider a Sun-like star and a single planet with the same characteristics as Jupiter. That is, $m_{B}=1.0 \times 10^{-3} m_{A}$ and the planet has an orbital radius of $8.0 \times 10^{11} \mathrm{m} .$ Determine the radius $r_{A}$ of the star's orbit about the system's CM.(c) When viewed from Earth, the distant system appears to wobble over a distance of 2$r_{A} .$ If astronomers are able to detect angular displacements $\theta$ of about 1 milliarcsec $\left(1$ arcsec $=\frac{1}{3600}$ of a degree), from what \right. distance $d$ (in light-years) can the star's wobble be detected $\left(1 \mathrm{ly}=9.46 \times 10^{15} \mathrm{m}\right) ?(d)$ The star nearest to our Sun is about 4 $\mathrm{ly}$ away. Assuming stars are uniformly distributed throughout our region of the Milky Way Galaxy, about how many stars can this technique be applied to in the search for extrasolar planetary systems?

Farhanul Hasan

Numerade Educator

Problem 113

Suppose two asteroids strike head on. Asteroid A $\left(m_{\mathrm{A}}=7.5 \times 10^{12} \mathrm{kg}\right)$ has velocity 3.3 $\mathrm{km} / \mathrm{s}$ before the collision, and asteroid $\quad \mathrm{B} \quad\left(m_{\mathrm{B}}=1.45 \times 10^{13} \mathrm{kg}\right)$ has velocity 1.4 $\mathrm{km} / \mathrm{s}$ before the collision in the opposite direction. If the asteroids stick together, what is the velocity (magnitude and direction) of the new asteroid after the collision?

Bruce Edelman

Numerade Educator

Problem 114

(III) A particle of mass $m_{\mathrm{A}}$ traveling with speed $v_{\mathrm{A}}$ collides elastically head-on with a stationary particle of smaller mass $m_{\mathrm{B}}$ . $(a)$ Show that the speed of $m_{\mathrm{B}}$ after the collision is
$v_{\mathrm{B}}^{\prime}=\frac{2 v_{\mathrm{A}}}{1+m_{\mathrm{B}} / m_{\mathrm{A}}}$
(b) Consider now a third particle of mass $m_{\mathrm{C}}$ at rest between $m_{\mathrm{A}}$ and $m_{\mathrm{B}}$ so that $m_{\mathrm{A}}$ first collides head on with $m_{\mathrm{C}}$ and then $m_{\mathrm{C}}$ collides head on with $m_{\mathrm{B}} .$ Both collisions are elastic. Show that in this case,
$v_{\mathrm{B}}^{\prime}=4 v_{\mathrm{A}} \frac{m_{\mathrm{C}} m_{\mathrm{A}}}{\left(m_{\mathrm{C}}+m_{\mathrm{A}}\right)\left(m_{\mathrm{B}}+m_{\mathrm{C}}\right)}$
(c) From the result of part $(b),$ show that for maximum $v_{B}^{\prime}, m_{C}=\sqrt{m_{A} m_{B}} \cdot(d)$ Assume $m_{B}=2.0 \mathrm{kg}$ $m_{A}=18.0 \mathrm{kg}$ and $v_{\mathrm{A}}=2.0 \mathrm{m} / \mathrm{s} .$ Use a spreadsheet to calculate and graph the values of $v_{\mathrm{B}}^{\prime}$ from $m_{\mathrm{C}}=0.0 \mathrm{kg}$ to $m_{\mathrm{C}}=50.0 \mathrm{kg}$ in steps of 1.0 $\mathrm{kg}$ . For what value of $m_{\mathrm{C}}$ is the value of $v_{\mathrm{B}}^{\prime}$ maximum? Does your numerical result agree with your result in part $(c) ?$

Farhanul Hasan

Numerade Educator