Fundamentals of Physics, Volume 2

David Halliday & Robert Resnick & Jearl Walker

Chapter 37

Relativity - all with Video Answers

Educators

Chapter Questions

Problem 1

The mean lifetime of stationary muons is measured to be $2.2000 \mu \mathrm{s}$. The mean lifetime of high-speed muons in a burst of cosmic rays observed from Earth is measured to be $16.000 \mu \mathrm{s}$. To five significant figures, what is the speed parameter $\beta$ of these cosmic-ray muons relative to Earth?

Suzanne W.

Suzanne W.

Numerade Educator

Problem 2

To eight significant figures, what is speed parameter $\beta$ if the Lorentz factor $\gamma$ is (a) 1.0100000 , (b) 10.000000 , (c) 100.00000 , and (d) 1000.0000 ?

Salamat Ali

Numerade Educator

Problem 3

You wish to make a round trip from Earth in a spaceship, traveling at constant speed in a straight line for exactly 6 months (as you measure the time interval) and then returning at the same constant speed. You wish further, on your return, to find Earth as it will be exactly 1000 years in the future. (a) To eight significant figures, at what speed parameter $\beta$ must you travel? (b) Does it matter whether you travel in a straight line on your journey?

Salamat Ali

Numerade Educator

Problem 4

(Come) back to the future. Suppose that a father is $20.00 \mathrm{y}$ older than his daughter. He wants to travel outward from Earth for $2.000 \mathrm{y}$ and then back for another $2.000 \mathrm{y}$ (both intervals as he measures them) such that he is then $20.00 \mathrm{y}$ younger than his daughter. What constant speed parameter $\beta$ (relative to Earth) is required?

Salamat Ali

Numerade Educator

Problem 5

An unstable high-energy particle enters a detector and leaves a track of length $1.05 \mathrm{~mm}$ before it decays. Its speed relative to the detector was $0.992 c$. What is its proper lifetime? That is, how long would the particle have lasted before decay had it been at rest with respect to the detector?

Suzanne W.

Numerade Educator

Problem 6

Reference frame $S^{\prime}$ is to pass reference frame $S$ at speed $v$ along the common direction of the $x^{\prime}$ and $x$ axes, as in Fig. 37.3.1. An observer who rides along with frame $S^{\prime}$ is to count off a certain time interval on his wristwatch. The corresponding time interval $\Delta t$ is to be measured by an observer in frame $S$. Figure 37.8 gives $\Delta t$ versus speed parameter $\beta$ for a range of values
for $\beta$. The vertical axis scale is set by $\Delta t_\alpha=14.0 \mathrm{~s}$. What is interval $\Delta t$ if $v=0.98 c$ ?
( FIGURE CAN'T COPY )

Salamat Ali

Numerade Educator

Problem 7

The premise of the Planet of the Apes movies and book is that hibernating astronauts travel far into Earth's future, to a time when human civilization has been replaced by an ape civilization. Considering only special relativity, determine how far into Earth's future the astronauts would travel if they slept for $120 \mathrm{y}$ while traveling relative to Earth with a speed of $0.9990 \mathrm{c}$, first outward from Earth and then back again.

Salamat Ali

Numerade Educator

Problem 8

An electron of $\beta=0.999987$ moves along the axis of an evacuated tube that has a length of $3.00 \mathrm{~m}$ as measured by a
laboratory observer $S$ at rest relative to the tube. An observer $S^{\prime}$ who is at rest relative to the electron, however, would see this tube moving with speed $v(=\beta c)$. What length would observer $S^{\prime}$ measure for the tube?

Salamat Ali

Numerade Educator

Problem 9

A spaceship of rest length $130 \mathrm{~m}$ races past a timing station at a speed of $0.740 c$. (a) What is the length of the spaceship as measured by the timing station? (b) What time interval will the station clock record between the passage of the front and back ends of the ship?

Salamat Ali

Numerade Educator

Problem 10

A meter stick in frame $S^{\prime}$ makes an angle of $30^{\circ}$ with the $x^{\prime}$ axis. If that frame moves parallel to the $x$ axis of frame $S$ with speed $0.90 \mathrm{c}$ relative to frame $S$, what is the length of the stick as measured from $S$ ?

Salamat Ali

Numerade Educator

Problem 11

A rod lies parallel to the $x$ axis of reference frame $S$, moving along this axis at a speed of $0.630 \mathrm{c}$. Its rest length is $1.70 \mathrm{~m}$. What will be its measured length in frame $S$ ?

Salamat Ali

Numerade Educator

Problem 12

The length of a spaceship is measured to be exactly half its rest length. (a) To three significant figures, what is the speed parameter $\beta$ of the spaceship relative to the observer's frame?
(b) By what factor do the spaceship's clocks run slow relative to clocks in the observer's frame?

Salamat Ali

Numerade Educator

Problem 13

A space traveler takes off from Earth and moves at speed $0.9900 \mathrm{c}$ toward the star Vega, which is 26.00 ly distant. How much time will have elapsed by Earth clocks (a) when the traveler reaches Vega and (b) when Earth observers receive word from the traveler that she has arrived? (c) How much older will Earth observers calculate the traveler to be (measured from her frame) when she reaches Vega than she was when she started the trip?

Salamat Ali

Numerade Educator

Problem 14

A rod is to move at constant speed $v$ along the $x$ axis of reference frame $S$, with the rod's length parallel to that axis. An observer in frame $S$ is to measure the length $L$ of the rod. Figure 37.9 gives length $L$ versus speed parameter $\beta$ for a range of values for $\beta$. The vertical axis scale is set by $L_\mu=1.00 \mathrm{~m}$. What is $L$ if $v=0.95 c$ ?
( FIGURE CAN'T COPY )

Suzanne W.

Numerade Educator

Problem 15

The center of our Milky Way Galaxy is about 23000 ly away. (a) To eight significant figures, at what constant speed parameter would you need to travel exactly $23000 \mathrm{ly}$ (measured in the Galaxy frame) in exactly $30 \mathrm{y}$ (measured in your frame)? (b) Measured in your frame and in light-years, what length of the Galaxy would pass by you during the trip?

Salamat Ali

Numerade Educator

Problem 16

Observer $S$ reports that an event occurred on the $x$ axis of his reference frame at $x=3.00 \times 10^8 \mathrm{~m}$ at time $t=2.50 \mathrm{~s}$. Observer $S^{\prime}$ and her frame are moving in the positive direction of the $x$ axis at a speed of $0.400 \mathrm{c}$. Further, $x=x^{\prime}=0$ at $t=t^{\prime}=0$. What are the (a) spatial and (b) temporal coordinate of the
event according to $S^{\prime}$ ? If $S^{\prime}$ were, instead, moving in the negative direction of the $x$ axis, what would be the (c) spatial and (d) temporal coordinate of the event according to $S^{\prime}$ ?

Salamat Ali

Numerade Educator

Problem 17

In Fig. 37.3.1, the origins of the two frames coincide at $t=t^{\prime}=0$ and the relative speed is $0.950 \mathrm{c}$. Two micrometeorites collide at coordinates $x=100 \mathrm{~km}$ and $t=200 \mu \mathrm{s}$ according to an observer in frame $S$. What are the (a) spatial and (b) temporal coordinate of the collision according to an observer in frame $S^{\prime \prime}$ ?

Prabhu Ramji

Numerade Educator

Problem 18

Inertial frame $S^{\prime}$ moves at a speed of $0.60 c$ with respect to frame $S$ (Fig. 37.3.1). Further, $x=x^{\prime}=0$ at $t=t^{\prime}=0$. Two events are recorded. In frame $S$, event 1 occurs at the origin at $t=0$ and event 2 occurs on the $x$ axis at $x=3.0 \mathrm{~km}$ at $t=4.0 \mu \mathrm{s}$. According to observer $S^{\prime}$, what is the time of (a) event 1 and (b) event 2 ? (c) Do the two observers see the same sequence or the reverse?

Raj Bala

Numerade Educator

Problem 19

An experimenter arranges to trigger two flashbulbs simultaneously, producing a big flash located at the origin of his reference frame and a small flash at $x=30.0 \mathrm{~km}$. An observer moving at a speed of $0.250 c$ in the positive direction of $x$ also views the flashes. (a) What is the time interval between them according to her? (b) Which flash does she say occurs first?

Salamat Ali

Numerade Educator

Problem 20

As in Fig. 37.3.1, reference frame $S^{\prime}$ passes reference frame $S$ with a certain velocity. Events 1 and 2 are to have a certain temporal separation $\Delta t^{\prime}$ according to the $S^{\prime}$ observer. However, their spatial separation $\Delta x^{\prime}$ according to that observer has not been set yet. Figure 37.10 gives their temporal separation $\Delta t$ according to the $S$ observer as a function of $\Delta x^{\prime}$ for a range of $\Delta x^{\prime}$ values. The vertical axis scale is set by $\Delta t_a=6.00 \mu \mathrm{s}$. What is $\Delta t^{\prime \prime}$ ?
( FIGURE CAN'T COPY )

Raj Bala

Numerade Educator

Problem 21

Relativistic reversal of events. Figures $37.11 a$ and $b$ show the (usual) situation in which a primed reference frame passes an unprimed reference frame, in the common positive direction of the $x$ and $x^{\prime}$ axes, at a constant relative velocity of magnitude $v$. We are at rest in the unprimed frame; Bullwinkle, an astute student of relativity in spite of his cartoon upbringing, is at rest in the primed frame. The figures also indicate events $A$ and $B$ that occur at the following spacetime coordinates as measured in our unprimed frame and in Bullwinkle's primed frame:
$$
\begin{array}{ccc}
\text { Event } & \text { Unprimed } & \text { Primed } \\
A & \left(x_A, t_A\right) & \left(x_A^{\prime}, t_A^{\prime}\right) \\
B & \left(x_B, t_B\right) & \left(x_B^{\prime}, t_B^{\prime}\right)
\end{array}
$$

In our frame, event $A$ occurs before event $B$, with temporal separation $\Delta t=t_B-t_A=1.00 \mu \mathrm{s}$ and spatial separation $\Delta x=$ $x_B-x_A=400 \mathrm{~m}$. Let $\Delta t^{\prime}$ be the temporal separation of the events according to Bullwinkle. (a) Find an expression for $\Delta t^{\prime}$ in terms of the speed parameter $\beta(=v / c)$ and the given data. Graph $\Delta t^{\prime}$ versus $\beta$ for the following two ranges of $\beta$ :
(b) 0 to $0.01 \quad(v$ is low, from 0 to $0.01 c)$
(c) 0.1 to $1 \quad$ ( $v$ is high, from $0.1 c$ to the limit $c$ )
(d) At what value of $\beta$ is $\Delta t^{\prime}=0$ ? For what range of $\beta$ is the sequence of events $A$ and $B$ according to Bullwinkle (e) the
same as ours and (f) the reverse of ours? (g) Can event $A$ cause event $B$, or vice versa? Explain.
A ( FIGURE CAN'T COPY )
B ( FIGURE CAN'T COPY )

Raj Bala

Numerade Educator

Problem 22

CALC For the passing reference frames in Fig. 37.11, events $A$ and $B$ occur at the following spacetime coordinates: according to the unprimed frame, $\left(x_A, t_A\right)$ and $\left(x_B, t_B\right)$; according to the primed frame, $\left(x_A^{\prime}, t_A^{\prime}\right)$ and $\left(x_B^{\prime}, t_B^{\prime}\right)$. In the unprimed frame, $\Delta t=t_B-t_A=1.00 \mu \mathrm{s}$ and $\Delta x=x_B-x_A=400 \mathrm{~m}$. (a) Find an expression for $\Delta x^{\prime}$ in terms of the speed parameter $\beta$ and the given data. Graph $\Delta x^{\prime}$ versus $\beta$ for two ranges of $\beta$ : (b) 0 to 0.01 and (c) 0.1 to 1 . (d) At what value of $\beta$ is $\Delta x^{\prime}$ minimum, and (e) what is that minimum?

Salamat Ali

Numerade Educator

Problem 23

A clock moves along an $x$ axis at a speed of $0.600 c$ and reads zero as it passes the origin of the axis. (a) Calculate the clock's Lorentz factor. (b) What time does the clock read as it passes $x=180 \mathrm{~m}$ ?

Salamat Ali

Numerade Educator

Problem 24

Bullwinkle in reference frame $S^{\prime}$ passes you in reference frame $S$ along the common direction of the $x^{\prime}$ and $x$ axes, as in Fig. 37.3.1. He carries three meter sticks: meter stick 1 is parallel to the $x^{\prime}$ axis, meter stick 2 is parallel to the $y^{\prime}$ axis, and meter stick 3 is parallel to the $z^{\prime}$ axis. On his wristwatch he counts off $15.0 \mathrm{~s}$, which takes $30.0 \mathrm{~s}$ according to you. Two events occur during his passage. According to you, event 1 occurs at $x_1=33.0 \mathrm{~m}$ and $t_1=22.0 \mathrm{~ns}$, and event 2 occurs at $x_2=53.0 \mathrm{~m}$ and $t_2=62.0 \mathrm{~ns}$. According to your measurements, what is the length of (a) meter stick 1, (b) meter stick 2, and (c) meter stick 3? According to Bullwinkle, what are (d) the spatial separation and (e) the temporal separation between events 1 and 2 , and (f) which event occurs first?

Suzanne W.

Numerade Educator

Problem 25

In Fig. 37.3.1, observer $S$ detects two flashes of light. A big flash occurs at $x_1=1200 \mathrm{~m}$ and, $5.00 \mu \mathrm{s}$ later, a small flash occurs at $x_2=480 \mathrm{~m}$. As detected by observer $S^{\prime}$, the two flashes occur at a single coordinate $x^{\prime}$. (a) What is the speed parameter of $S^{\prime}$, and (b) is $S^{\prime}$ moving in the positive or negative direction of the $x$ axis? To $S^{\prime}$, (c) which flash occurs first and (d) what is the time interval between the flashes?

Raj Bala

Numerade Educator

Problem 26

In Fig. 37.3.1, observer $S$ detects two flashes of light. A big flash occurs at $x_1=1200 \mathrm{~m}$ and, slightly later, a small flash occurs at $x_2=480 \mathrm{~m}$. The time interval between the flashes is $\Delta t=t_2-t_1$. What is the smallest value of $\Delta t$ for which observer $S^{\prime}$ will determine that the two flashes occur at the same $x^{\prime}$ coordinate?

Narayan Hari

Numerade Educator

Problem 27

A particle moves along the $x^{\prime}$ axis of frame $S^{\prime}$ with velocity $0.40 c$. Frame $S^{\prime}$ moves with velocity $0.60 c$ with respect to frame $S$. What is the velocity of the particle with respect to frame $S$ ?

Suzanne W.

Numerade Educator

Problem 28

In Fig. 37.4.1, frame $S^{\prime}$ moves relative to frame $S$ with velocity $0.62 c i$ while a particle moves parallel to the common $x$ and $x^{\prime}$ axes. An observer attached to frame $S^{\prime}$ measures the particle's velocity to be $0.47 \hat{c}$. In terms of $c$, what is the particle's velocity as measured by an observer attached to frame $S$ according to the (a) relativistic and (b) classical velocity transformation? Suppose, instead, that the $S^{\prime}$ measure of the particle's velocity is $-0.47 c \hat{i}$. What velocity does the observer in $S$ now measure according to the (c) relativistic and (d) classical velocity transformation?

Suzanne W.

Numerade Educator

Problem 29

Galaxy $\mathrm{A}$ is reported to be receding from us with a speed of $0.35 \mathrm{c}$. Galaxy $\mathrm{B}$, located in precisely the opposite direction, is also found to be receding from us at this same speed. What multiple of $c$ gives the recessional speed an observer on Galaxy A would find for (a) our galaxy and (b) Galaxy B?

Salamat Ali

Numerade Educator

Problem 30

EStellar system $Q_1$ moves away from us at a speed of $0.800 c$. Stellar system $Q_2$, which lies in the same direction in space but is closer to us, moves away from us at speed $0.400 \mathrm{c}$. What multiple of $c$ gives the speed of $Q_2$ as measured by an observer in the reference frame of $Q_1$ ?

Raj Bala

Numerade Educator

Problem 31

A spaceship whose rest length is $350 \mathrm{~m}$ has a speed of $0.82 c$ with respect to a certain reference frame. A micrometeorite, also with a speed of $0.82 c$ in this frame, passes the spaceship on an antiparallel track. How long does it take this object to pass the ship as measured on the ship?

Suzanne W.

Numerade Educator

Problem 32

In Fig. 37.12a, particle $P$ is to move parallel to the $x$ and $x^{\prime}$ axes of reference frames $S$ and $S^{\prime}$, at a certain velocity relative to frame $S$. Frame $S^{\prime}$ is to move parallel to the $x$ axis of frame $S$ at velocity $v$. Figure $37.12 b$ gives the velocity $u^{\prime}$ of the particle relative to frame $S^{\prime}$ for a range of values for $v$. The vertical axis scale is set by $u_\alpha^{\prime}=0.800 \mathrm{c}$. What value will $u^{\prime}$ have if (a) $v=0.90 c$ and (b) $v \rightarrow c$ ?
A ( FIGURE CAN'T COPY )
B( FIGURE CAN'T COPY )

Raj Bala

Numerade Educator

Problem 33

An armada of spaceships that is $1.00 \mathrm{ly}$ long (as measured in its rest frame) moves with speed $0.800 \mathrm{c}$ relative to a ground station in frame $S$. A messenger travels from the rear of the armada to the front with a speed of $0.950 \mathrm{c}$ relative to $S$. How long does the trip take as measured (a) in the rest frame of the messenger, (b) in the rest frame of the armada, and (c) by an observer in the ground frame $S$ ?

Salamat Ali

Numerade Educator

Problem 34

A sodium light source moves in a horizontal circle at a constant speed of $0.100 \mathrm{c}$ while emitting light at the proper wavelength of $\lambda_0=589.00 \mathrm{~nm}$. Wavelength $\lambda$ is measured for that
light by a detector fixed at the center of the circle. What is the wavelength shift $\lambda-\lambda_0$ ?

Salamat Ali

Numerade Educator

Problem 35

A spaceship, moving away from Earth at a speed of $0.900 \mathrm{c}$, reports back by transmitting at a frequency (measured in the spaceship frame) of $100 \mathrm{MHz}$. To what frequency must Earth receivers be tuned to receive the report?

Salamat Ali

Numerade Educator

Problem 36

Certain wavelengths in the light from a galaxy in the constellation Virgo are observed to be $0.4 \%$ longer than the corresponding light from Earth sources. (a) What is the radial speed of this galaxy with respect to Earth? (b) Is the galaxy approaching or receding from Earth?

Salamat Ali

Numerade Educator

Problem 37

Assuming that Eq. 37.5 .6 holds, find how fast you would have to go through a red light to have it appear green. Take $620 \mathrm{~nm}$ as the wavelength of red light and $540 \mathrm{~nm}$ as the wavelength of green light.

Salamat Ali

Numerade Educator

Problem 38

Figure 37.13 is a graph of intensity versus wavelength for light reaching Earth from galaxy NGC 7319, which is about $3 \times 10^8$ ly away. The most intense light is emitted by the oxygen in NGC 7319. In a laboratory that emission is at wavelength $\lambda=513 \mathrm{~nm}$, but in the light from NGC 7319 it has been shifted to $525 \mathrm{~nm}$ due to the Doppler effect (all the emissions from NGC 7319 have been shifted). (a) What is the radial speed of NGC 7319 relative to Earth? (b) Is the relative motion toward or away from our planet?
( FIGURE CAN'T COPY )

Salamat Ali

Numerade Educator

Problem 39

A spaceship is moving away from Earth at speed $0.20 \mathrm{c}$. A source on the rear of the ship emits light at wavelength $450 \mathrm{~nm}$ according to someone on the ship. What (a) wavelength and (b) color (blue, green, yellow, or red) are detected by someone on Earth watching the ship?

Salamat Ali

Numerade Educator

Problem 40

How much work must be done to increase the speed of an electron from rest to (a) $0.500 c$, (b) $0.990 c$, and (c) $0.9990 c$ ?

Raj Bala

Numerade Educator

Problem 41

The mass of an electron is $9.10938188 \times 10^{-31} \mathrm{~kg}$. To six significant figures, find (a) $\gamma$ and (b) $\beta$ for an electron with kinetic energy $K=100.000 \mathrm{MeV}$.

Salamat Ali

Numerade Educator

Problem 42

What is the minimum energy that is required to break a nucleus of ${ }^{12} \mathrm{C}$ (of mass $11.99671 \mathrm{u}$ ) into three nuclei of ${ }^4 \mathrm{He}$ (of mass $4.00151 \mathrm{u}$ each)?

Salamat Ali

Numerade Educator

Problem 43

How much work must be done to increase the speed of an electron (a) from $0.18 c$ to $0.19 c$ and (b) from $0.98 c$ to $0.99 c$ ?

Suzanne W.

Numerade Educator

Problem 44

In the reaction $\mathrm{p}+{ }^{19} \mathrm{~F} \rightarrow \alpha+{ }^{16} \mathrm{O}$, the masses are
$$
\begin{array}{ll}
m(\mathrm{p})=1.007825 \mathrm{u}, & m(\alpha)=4.002603 \mathrm{u}, \\
m(\mathrm{~F})=18.998405 \mathrm{u}, & m(\mathrm{O})=15.994915 \mathrm{u} .
\end{array}
$$

Calculate the $Q$ of the reaction from these data.

Salamat Ali

Numerade Educator

Problem 45

In a high-energy collision between a cosmic-ray particle and a particle near the top of Earth's atmosphere, $120 \mathrm{~km}$ above sea level, a pion is created. The pion has a total energy $E$ of $1.35 \times 10^5 \mathrm{MeV}$ and is traveling vertically downward. In the pion's rest frame, the pion decays $35.0 \mathrm{~ns}$ after its creation. At what altitude above sea level, as measured from Earth's reference frame, does the decay occur? The rest energy of a pion is $139.6 \mathrm{MeV}$.

Raj Bala

Numerade Educator

Problem 46

(a) If $m$ is a particle's mass, $p$ is its momentum magnitude, and $K$ is its kinetic energy, show that
$$
m=\frac{(p c)-K^2}{2 K c^2} .
$$
(b) For low particle speeds, show that the right side of the equation reduces to $m$. (c) If a particle has $K=55.0 \mathrm{MeV}$ when $p=$ $121 \mathrm{MeV} / \mathrm{c}$, what is the ratio $\mathrm{m} / \mathrm{m}_e$ of its mass to the electron mass?

Prabhu Ramji

Numerade Educator

Problem 47

A 5.00-grain aspirin tablet has a mass of $320 \mathrm{mg}$. For how many kilometers would the energy equivalent of this mass power an automobile? Assume $12.75 \mathrm{~km} / \mathrm{L}$ and a heat of combustion of $3.65 \times 10^7 \mathrm{~J} / \mathrm{L}$ for the gasoline used in the automobile.

Suzanne W.

Numerade Educator

Problem 48

The mass of a muon is 207 times the electron mass; the average lifetime of muons at rest is $2.20 \mu \mathrm{s}$. In a certain experiment, muons moving through a laboratory are measured to have an average lifetime of $6.90 \mu \mathrm{s}$. For the moving muons, what are (a) $\beta$, (b) $K$, and (c) $p$ (in $\mathrm{MeV} / c)$ ?

Salamat Ali

Numerade Educator

Problem 49

As you read this page (on paper or monitor screen), a cosmic ray proton passes along the left-right width of the page with relative speed $v$ and a total energy of $14.24 \mathrm{~nJ}$. According to your measurements, that left-right width is $21.0 \mathrm{~cm}$. (a) What is the width according to the proton's reference frame? How much time did the passage take according to (b) your frame and (c) the proton's frame?

Salamat Ali

Numerade Educator

Problem 50

To four significant figures, find the following when the kinetic energy is $10.00 \mathrm{MeV}$ : (a) $\gamma$ and (b) $\beta$ for an electron ( $E_0=$ $0.510998 \mathrm{MeV})$, (c) $\gamma$ and (d) $\beta$ for a proton $\left(E_0=938.272 \mathrm{MeV}\right)$, and (c) $\gamma$ and (f) $\beta$ for an $\alpha$ particle ( $E_0=3727.40 \mathrm{MeV}$ ).

Salamat Ali

Numerade Educator

Problem 51

What must be the momentum of a particle with mass $m$ so that the total energy of the particle is 3.00 times its rest energy?

Salamat Ali

Numerade Educator

Problem 52

Apply the binomial theorem (Appendix E) to the last part of Eq. 37.6 .13 for the kinetic energy of a particle. (a) Retain the first two terms of the expansion to show the kinetic energy in the form
$$
K=(\text { first term })+(\text { second term }) .
$$

The first term is the classical expression for kinetic energy. The second term is the first-order correction to the classical expression. Assume the particle is an electron. If its speed $v$ is $c / 20$, what is the value of (b) the classical expression and (c) the first-order correction? If the electron's speed is $0.80 c$, what is the value of (d) the classical expression and (e) the first-order correction? (f) At what speed parameter $\beta$ does the first-order correction become $10 \%$ or greater of the classical expression?

Raj Bala

Numerade Educator

Problem 53

In Module 28.4, we showed that a particle of charge $q$ and mass $m$ will move in a circle of radius $r=m v /|q| B$ when its velocity $\vec{v}$ is perpendicular to a uniform magnetic field $\vec{B}$. We also found that the period $T$ of the motion is independent of speed $v$. These two results are approximately correct if $v \& c$. For relativistic speeds, we must use the correct equation for the radius:
$$
r=\frac{p}{|q| B}=\frac{\gamma m v}{|q| B}
$$
(a) Using this equation and the definition of period $(T=2 \pi r / v)$, find the correct expression for the period. (b) Is $T$ independent of $v$ ? If a $10.0 \mathrm{MeV}$ electron moves in a circular path in a uniform magnetic field of magnitude $2.20 \mathrm{~T}$, what are (c) the radius according to Chapter 28 , (d) the correct radius, (e) the period according to Chapter 28 , and (f) the correct period?

Suzanne W.

Numerade Educator

Problem 54

(6. What is $\beta$ for a particle with (a) $K=2.00 E_0$ and (b) $E=2.00 E_0$ ?

Salamat Ali

Numerade Educator

Problem 55

A certain particle of mass $m$ has momentum of magnitude $m c$. What are (a) $\beta$, (b) $\gamma$, and (c) the ratio $K / E_0$ ?

Suzanne W.

Numerade Educator

Problem 56

(a) The energy released in the explosion of $1.00 \mathrm{~mol}$ of TNT is $3.40 \mathrm{MJ}$. The molar mass of TNT is $0.227 \mathrm{~kg} / \mathrm{mol}$. What weight of TNT is needed for an explosive release of $1.80 \times$ $10^{14} \mathrm{~J}$ ? (b) Can you carry that weight in a backpack, or is a truck or train required? (c) Suppose that in an explosion of a fission bomb, $0.080 \%$ of the fissionable mass is converted to released energy. What weight of fissionable material is needed for an explosive release of $1.80 \times 10^{14} \mathrm{~J}$ ? (d) Can you carry that weight in a backpack, or is a truck or train required?

Salamat Ali

Numerade Educator

Problem 57

CALC Quasars are thought to be the nuclei of active galaxies in the early stages of their formation. A typical quasar radiates energy at the rate of $10^{41} \mathrm{~W}$. At what rate is the mass of this quasar being reduced to supply this energy? Express your answer in solar mass units per year, where one solar mass unit $\left(1 \mathrm{smu}=2.0 \times 10^{30} \mathrm{~kg}\right)$ is the mass of our Sun.

Salamat Ali

Numerade Educator

Problem 58

The mass of an electron is $9.10938188 \times 10^{-31} \mathrm{~kg}$. To eight significant figures, find the following for the given electron kinetic energy: (a) $\gamma$ and (b) $\beta$ for $K=1.0000000 \mathrm{keV}$, (c) $\gamma$ and (d) $\beta$ for $K=1.0000000 \mathrm{MeV}$, and then (e) $\gamma$ and (f) $\beta$ for $K=1.0000000 \mathrm{GeV}$.

Salamat Ali

Numerade Educator

Problem 59

An alpha particle with kinetic energy $7.70 \mathrm{MeV}$ collides with an ${ }^{14} \mathrm{~N}$ nucleus at rest, and the two transform into an ${ }^{17} \mathrm{O}$ nucleus and a proton. The proton is emitted at $90^{\circ}$ to the direction of the incident alpha particle and has a kinetic energy of $4.44 \mathrm{MeV}$. The masses of the various particles are alpha particle, $4.00260 \mathrm{u} ;{ }^{14} \mathrm{~N}, 14.00307 \mathrm{u}$; proton, $1.007825 \mathrm{u}$; and ${ }^{17} \mathrm{O}$, $16.99914 \mathrm{u}$. In MeV, what are (a) the kinetic energy of the oxygen nucleus and (b) the $Q$ of the reaction? (Hint: The speeds of the particles are much less than $c$.)

Keshav Singh

Numerade Educator

Problem 60

Temporal separation between two events. Events $A$ and $B$ occur with the following spacetime coordinates in the reference frames of Fig. 37.11: according to the unprimed frame, $\left(x_A, t_A\right)$ and $\left(x_B, t_B\right)$; according to the primed frame, $\left(x_A^{\prime}, t_A^{\prime}\right)$ and $\left(x_B^{\prime}, t_B^{\prime}\right)$. In the unprimed frame, $\Delta t=t_B-t_A=1.00 \mu \mathrm{s}$ and $\Delta x=$ $x_B-x_A=240 \mathrm{~m}$. (a) Find an expression for $\Delta t^{\prime}$ in terms of the speed parameter $\beta$ and the given data. Graph $\Delta t^{\prime}$ versus $\beta$ for the following two ranges of $\beta$ : (b) 0 to 0.01 and (c) 0.1 to 1 . (d) At what value of $\beta$ is $\Delta t^{\prime}$ minimum and (e) what is that minimum? (f) Can one of these events cause the other? Explain.

Salamat Ali

Numerade Educator

Problem 61

Spatial separation between two events. For the passing reference frames of Fig. 37.11, events $A$ and $B$ occur with the following spacetime coordinates: according to the unprimed frame, $\left(x_A, t_A\right)$ and $\left(x_B, t_B\right)$; according to the primed frame, $\left(x_A^{\prime}, t_A^{\prime}\right)$ and $\left(x_B^{\prime}, t_B^{\prime}\right)$. In the unprimedframe, $\Delta t=t_B-t_A=1.00 \mu \mathrm{s}$ and $\Delta x=x_B-x_A=240 \mathrm{~m}$. (a) Find an expression for $\Delta x^{\prime}$ in terms of the speed parameter $\beta$ and the given data. Graph $\Delta x^{\prime}$ versus $\beta$ for two ranges of $\beta$ : (b) 0 to 0.01 and (c) 0.1 to 1 . (d) At what value of $\beta$ is $\Delta x^{\prime}=0$ ?

Salamat Ali

Numerade Educator

Problem 62

In Fig. 37.14a, particle $P$ is to move parallel to the $x$ and $x^{\prime}$ axes of reference frames $S$ and $S^{\prime}$, at a certain velocity relative to frame $S$. Frame $S^{\prime}$ is to move parallel to the $x$ axis of frame $S$ at velocity $v$. Figure $37.14 b$ gives the velocity $u^{\prime}$ of the particle relative to frame $S^{\prime}$ for a range of values for $v$. The vertical axis scale is set by $u_\alpha^{\prime}=-0.800 \mathrm{c}$. What value will $u^{\prime}$ have if (a) $v=0.80 c$ and (b) $v \rightarrow c$ ?
A ( FIGURE CAN'T COPY )
B ( FIGURE CAN'T COPY )

Salamat Ali

Numerade Educator

Problem 63

Superluminal jets. Figure $37.15 a$ shows the path taken by a knot in a jet of ionized gas that has been expelled from a galaxy. The knot travels at constant velocity $\vec{v}$ at angle $\theta$ from the direction of Earth. The knot occasionally emits a burst of light, which is eventually detected on Earth. Two bursts are indicated in Fig. 37.15a, separated by time $t$ as measured in a stationary frame near the bursts. The bursts are shown in Fig. $37.15 b$ as if
A ( FIGURE CAN'T COPY )
B ( FIGURE CAN'T COPY )
they were photographed on the same piece of film, first when light from bust 1 an ived on Eanth and then later when light fron burst 2 arrived. The apparent distance $D_{\text {app }}$ traveled by the knot between the two bursts is the distance across an Earth-observer's view of the knot's path. The apparent time $T_{\text {app }}$ between the bursts is the difference in the arrival times of the light from them. The apparent speed of the knot is then $V_{\text {app }}=D_{\text {app }} / T_{\text {app }}$. In terms of $v, t$, and $\theta$, what are (a) $D_{\text {app }}$ and (b) $T_{\text {app }}$ ? (c) Evaluate $V_{\text {app }}$ for $v=0.980 \mathrm{c}$ and $\theta=30.0^{\circ}$. When superluminal (faster than light) jets were first observed, they seemed to defy special relativity-at least until the correct geometry (Fig. 37.15a) was understood.

Suzanne W.

Numerade Educator

Problem 64

Reference frame $S^{\prime}$ passes reference frame $S$ with a certain velocity as in Fig. 37.3.1. Events 1 and 2 are to have a certain spatial separation $\Delta x^{\prime}$ according to the $S^{\prime}$ observer. However, their temporal separation $\Delta t^{\prime}$ according to that observer has not been set yet. Figure 37.16 gives their spatial separation $\Delta x$ according to the $S$ observer as a function of $\Delta t^{\prime}$ for a range of $\Delta t^{\prime}$ values. The vertical axis scale is set by $\Delta x_a=$ $10.0 \mathrm{~m}$. What is $\Delta x$ ?
( FIGURE CAN'T COPY )

Salamat Ali

Numerade Educator

Problem 65

Another approach to velocity transformations. In Fig. 37.17, reference frames $B$ and $C$ move past reference frame $A$ in the common direction of their $x$ axes. Represent the $x$ components of the velocities of one frame relative to another with a twoletter subscript. For example, $v_{A B}$ is the $x$ component of the velocity of $A$ relative to $B$. Similarly, represent the corresponding speed parameters with two-letter subscripts. For example, $\beta_{A B}\left(=v_{A B} / c\right)$ is the speed parameter corresponding to $v_{A B}$. (a) Show that
$$
\beta_{A C}=\frac{\beta_{A B}+\beta_{B C}}{1+\beta_{A B} \beta_{B C}} .
$$

Let $M_{A B}$ represent the ratio $\left(1-\beta_{A B}\right) /\left(1+\beta_{A B}\right)$, and let $M_{B C}$ and $M_{A C}$ represent similar ratios. (b) Show that the relation
$$
M_{A C}=M_{A B} M_{B C}
$$
is true by deriving the equation of part (a) from it.
( FIGURE CAN'T COPY )

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 66

Continuation of Problem 65. Use the result of part (b) in Problem 65 for the motion along a single axis in the following situation. Frame $A$ in Fig. 37.17 is attached to a particle that moves with velocity $+0.500 \mathrm{c}$ past frame $B$, which moves past frame $C$ with a velocity of $+0.500 \mathrm{c}$. What are (a) $M_{A C}$, (b) $\beta_{A C}$, and (c) the velocity of the particle relative to frame $C$ ?

Salamat Ali

Numerade Educator

Problem 67

Continuation of Problem 65. Let reference frame $C$ in Fig. 37.17 move past reference frame $D$ (not shown). (a) Show that
$$
M_{A D}=M_{A B} M_{B C} M_{C D}-
$$
(b) Now put this general result to work: Three particles move parallel to a single axis on which an observer is stationed. Let plus and minus signs indicate the directions of motion along that axis. Particle $A$ moves past particle $B$ at $\beta_{A B}=+0.20$. Particle $B$ moves past particle $C$ at $\beta_{B C}=-0.40$. Particle $C$ moves past observer $D$ at $\beta_{C D}=+0.60$. What is the velocity of particle $A$ relative to observer $D$ ? (The solution technique here is much faster than using Eq. 37.4.1.)

Suzanne W.

Numerade Educator

Problem 68

Figure 37.2 shows a ship (attached to reference frame $S$ ) passing us (standing in reference frame $S$ ) with velocity $\vec{v}=$ $0.950 \mathrm{ci}$. A proton is fired at speed $0.980 \mathrm{c}$ relative to the ship from the front of the ship to the rear. The proper length of the ship is $760 \mathrm{~m}$. What is the temporal separation between the time the proton is fired and the time it hits the rear wall of the ship according to (a) a passenger in the ship and (b) us? Suppose that, instead, the proton is fired from the rear to the front. What then is the temporal separation between the time it is fired and the time it hits the front wall according to (c) the passenger and (d) us?

Keshav Singh

Numerade Educator

Problem 69

The car-in-the-garage problem. Carman has just purchased the world's longest stretch limo, which has a proper length of $L_c=30.5 \mathrm{~m}$. In Fig. 37.18a, it is shown parked in front of a garage with a proper length of $L_g=6.00 \mathrm{~m}$. The garage has a front door (shown open) and a back door (shown closed). The limo is obviously longer than the garage. Still, Garageman, who owns the garage and knows something about relativistic length contraction, makes a bet with Carman that the limo can fit in the garage with both doors closed. Carman, who dropped his physics course before reaching special relativity, says such a thing, even in principle, is impossible.

To analyze Garageman's scheme, an $x_c$ axis is attached to the limo, with $x_c=0$ at the rear bumper, and an $x_g$ axis is attached to the garage, with $x_g=0$ at the (now open) front door. Then Carman is to drive the limo directly toward the front door at a velocity of $0.9980 \mathrm{c}$ (which is, of course, both technically and financially impossible). Carman is stationary in the $x_c$ reference frame; Garageman is stationary in the $x_g$ reference frame.

There are two events to consider. Event 1: When the rear bumper clears the front door, the front door is closed. Let the time of this event be zero to both Carman and Garageman: $t_{{ }^1}=t_{c 1}=0$. The event occurs at $x_c=x_g=0$. Figure $37.18 b$ shows event 1 according to the $x_g$ reference frame. Event 2 : When the front bumper reaches the back door, that door opens. Figure 37.18 $c$ shows event 2 according to the $x_g$ reference frame.
A ( FIGURE CAN'T COPY )
B ( FIGURE CAN'T COPY )
C ( FIGURE CAN'T COPY )
According to Garageman, (a) what is the length of the limo, and what are the spacetime coordinates (b) $x_{g 2}$ and (c) $t_{g 2}$ of event 2 ? (d) For how long is the limo temporarily "trapped" inside the garage with both doors shut? Now consider the situation from the $x_c$ reference frame, in which the garage comes racing past the limo at a velocity of -0.9980 c. According to Carman, (e) what is the length of the passing garage, what are the spacetime coordinates (f) $x_{c 2}$ and (g) $t_{c 2}$ of event 2 , (h) is the limo ever in the garage with both doors shut, and (i) which event occurs first? (j) Sketch events 1 and 2 as seen by Carman. (k) Are the events causally related; that is, does one of them cause the other? (1) Finally, who wins the bet?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 70

An airplane has rest length $40.0 \mathrm{~m}$ and speed $630 \mathrm{~m} / \mathrm{s}$. To a ground observer, (a) by what fraction is its length contracted and (b) how long is needed for its clocks to be $1.00 \mu \mathrm{s}$ slow?

Salamat Ali

Numerade Educator

Problem 71

To circle Earth in low orbit, a satellite must have a speed of about $2.7 \times 10^4 \mathrm{~km} / \mathrm{h}$. Suppose that two such satellites orbit Earth in opposite directions. (a) What is their relative speed as they pass, according to the classical Galilean velocity transformation equation? (b) What fractional error do you make in (a) by not using the (correct) relativistic transformation equation?

Salamat Ali

Numerade Educator

Problem 72

Find the speed parameter of a particle that takes $2.0 \mathrm{y}$ longer than light to travel a distance of $6.0 \mathrm{ly}$.

Salamat Ali

Numerade Educator

Problem 73

How much work is needed to accelerate a proton from a speed of $0.9850 c$ to a speed of $0.9860 c$ ?

Salamat Ali

Numerade Educator

Problem 74

A pion is created in the higher reaches of Earth's atmosphere when an incoming high-energy cosmic-ray particle collides with an atomic nucleus. A pion so formed descends toward Earth with a speed of $0.99 \mathrm{c}$. In a reference frame in which they are at rest, pions decay with an average life of $26 \mathrm{~ns}$. As measured in a frame fixed with respect to Earth, how far (on the average) will such a pion move through the atmosphere before it decays?

Salamat Ali

Numerade Educator

Problem 75

If we intercept an electron having total energy $1533 \mathrm{MeV}$ that came from Vega, which is $26 \mathrm{ly}$ from us, how far in light-years was the trip in the rest frame of the electron?

Salamat Ali

Numerade Educator

Problem 76

The total energy of a proton passing through a laboratory apparatus is $10.611 \mathrm{~nJ}$. What is its speed parameter $\beta$ ? Use the proton mass given in Appendix B under "Best Value," not the commonly remembered rounded number.

Salamat Ali

Numerade Educator

Problem 77

A spaceship at rest in a certain reference frame $S$ is given a speed increment of $0.50 \mathrm{c}$. Relative to its new rest frame, it is then given a further $0.50 \mathrm{c}$ increment. This process is continued until its speed with respect to its original frame $S$ exceeds 0.999 c. How many increments does this process require?

Salamat Ali

Numerade Educator

Problem 78

In the red shift of radiation from a distant galaxy, a certain radiation, known to have a wavelength of $434 \mathrm{~nm}$ when observed in the laboratory, has a wavelength of $462 \mathrm{~nm}$. (a) What is the radial speed of the galaxy relative to Earth? (b) Is the galaxy approaching or receding from Earth?

Salamat Ali

Numerade Educator

Problem 79

What is the momentum in MeV/c of an electron with a kinetic energy of $2.00 \mathrm{MeV}$ ?

Salamat Ali

Numerade Educator

Problem 80

The radius of Earth is $6370 \mathrm{~km}$, and its orbital speed about the Sun is $30 \mathrm{~km} / \mathrm{s}$. Suppose Earth moves past an observer at this speed. To the observer, by how much does Earth's diameter contract along the direction of motion?

Suzanne W.

Numerade Educator

Problem 81

A particle with mass $m$ has speed $c / 2$ relative to inertial frame $S$. The particle collides with an identical particle at rest relative to frame $S$. Relative to $S$, what is the speed of a frame $S^{\prime}$ in which the total momentum of these particles is zero? This frame is called the center of momentum frame.

Salamat Ali

Numerade Educator

Problem 82

An elementary particle produced in a laboratory experiment travels $0.230 \mathrm{~mm}$ through the lab at a relative speed of $0.960 c$ before it decays (becomes another particle). (a) What is the proper lifetime of the particle? (b) What is the distance the particle travels as measured from its rest frame?

Salamat Ali

Numerade Educator

Problem 83

What are (a) $K$, (b) $E$, and (c) $p$ (in $\mathrm{GeV} / c$ ) for a proton moving at speed $0.990 c$ ? What are (d) $K$, (e) $E$, and (f) $p$ (in $\mathrm{MeV} / c$ ) for an electron moving at speed $0.990 c$ ?

Salamat Ali

Numerade Educator

Problem 84

A radar transmitter $T$ is fixed to a reference frame $S$ that is moving to the right with speed $v$ relative to reference frame $S$ (Fig. 37.19). A mechanical timer (essentially a clock) in
frame $S^{\prime}$, having a period $\tau_0$ (measured in $S^{\prime}$ ), causes transmitter $T$ to emit timed radar pulses, which travel at the speed of light and are received by $R$, a receiver fixed in frame $S$. (a) What is the period $\tau$ of the timer as detected by observer $A$, who is fixed in frame $S$ ? (b) Show that at receiver $R$ the time interval between pulses arriving from $T$ is not $\tau$ or $\tau_0$, but
$$
\tau_R=\tau_0 \sqrt{\frac{c+v}{c-v}} .
$$
(c) Explain why receiver $R$ and observer $A$, who are in the same reference frame, measure a different period for the transmitter.
( FIGURE CAN'T COPY )

Suzanne W.

Numerade Educator

Problem 85

One cosmic-ray particle approaches Earth along Earth's north-south axis with a speed of $0.80 \mathrm{c}$ toward the geographic north pole, and another approaches with a speed of $0.60 \mathrm{c}$ toward the geographic south pole (Fig. 37.20). What is the relative speed of approach of one particle with respect to the other?
( FIGURE CAN'T COPY )

Salamat Ali

Numerade Educator

Problem 86

(a) How much energy is released in the explosion of a fission bomb containing $3.0 \mathrm{~kg}$ of fissionable material? Assume that $0.10 \%$ of the mass is converted to released energy. (b) What mass of TNT would have to explode to provide the same energy release? Assume that each mole of TNT liberates $3.4 \mathrm{MJ}$ of energy on exploding. The molecular
mass of TNT is $0.227 \mathrm{~kg} / \mathrm{mol}$. (c) For the same mass of explosive, what is the ratio of the energy released in a nuclear explosion to that released in a TNT explosion?

Salamat Ali

Numerade Educator

Problem 87

(a) What potential difference would accelerate an electron to speed $c$ according to classical physics? (b) With this potential difference, what speed would the electron actually attain?

Salamat Ali

Numerade Educator

Problem 88

A Foron cruiser moving directly toward a Reptulian scout ship fires a decoy toward the scout ship. Relative to the
scout ship, the speed of the decoy is $0.980 \mathrm{c}$ and the speed of the Foron cruiser is $0.900 \mathrm{c}$. What is the speed of the decoy relative to the cruiser?

Suzanne W.

Numerade Educator

Problem 89

In Fig. 37.21, three spaceships are in a chase. Relative to an $x$ axis in an inertial frame (say, Earth frame), their velocities are $v_A=0.900 c, v_B$, and $v_C=0.800 c$. (a) What value of $v_B$ is required such that ships $A$ and $C$ approach ship $B$ with the same speed relative to ship $B$, and (b) what is that relative speed?
( FIGURE CAN'T COPY )

Salamat Ali

Numerade Educator

Problem 90

Space cruisers $A$ and $B$ are moving parallel to the positive direction of an $x$ axis. Cruiser $A$ is faster, with a relative speed of $v=0.900 \mathrm{c}$, and has a proper length of $L=200 \mathrm{~m}$. According to the pilot of $A$, at the instant $(t=0)$ the tails of the cruisers are aligned, the noses are also. According to the pilot of $B$, how much later are the noses aligned?

Salamat Ali

Numerade Educator

Problem 91

In Fig. 37.22, two cruisers fly toward a space station. Relative to the station, cruiser $A$ has speed $0.800 \mathrm{c}$. Relative to the station, what speed is required of cruiser $B$ such that its pilot sees $A$ and the station approach $B$ at the same speed?
( FIGURE CAN'T COPY )

Salamat Ali

Numerade Educator

Problem 92

A relativistic train of proper length $200 \mathrm{~m}$ approaches a tunnel of the same proper length, at a relative speed of $0.900 \mathrm{c}$. A paint bomb in the engine room is set to explode (and cover everyone with blue paint) when the front of the train passes the far end of the tunnel (event FF). However, when the rear car passes the near end of the tunnel (event RN), a device in that car is set to send a signal to the engine room to deactivate the bomb. Train view: (a) What is the tunnel length? (b) Which event occurs first, FF or RN? (c) What is the time between those events? (d) Does the paint bomb explode? Tunnel view: (e) What is the train length? (f) Which event occurs first? (g) What is the time between those events? (h) Does the paint bomb explode? If your answers to (d) and (h) differ, you need to explain the paradox, because either the engine room is covered with blue paint or not; you cannot have it both ways. If your answers are the same, you need to explain why.

Keshav Singh

Numerade Educator

Problem 93

Police radar. A police car equipped with a radar unit waits alongside a highway. The radar emits a beam of microwaves down the highway at frequency $f_0=24.125 \mathrm{GHz}$ (in the common $\mathrm{K}$ band used by police nationwide). When a vehicle travels toward the radar unit and through the beam, the microwaves reflect from the metal of the vehicle back to the radar unit. The unit can then determine the vehicle's speed from the beat frequency (see Module 17.6) between that incoming frequency and
the emitted frequency. If the beat frequency is $5000 \mathrm{~Hz}$, what is the vehicle's speed?

Suzanne W.

Numerade Educator

Problem 94

Time is short. You command a starship capable of traveling at nearly the speed of light. Beginning at Home Port in Fig. 37.23, you need to pick the route to Far Base that minimizes the travel time. The map gives the allowed routes as negotiated with the alien government governing the region, and each route between junction points is labeled with the Lorentz factor $\gamma$ that must be used along the route. In the rest frame of the junctions, successive junctions are separated by distance $L$ or $2 L$. Do not consider the time required for acceleration as $\gamma$ changes. (a) First, what length do you measure for a map distance of $L$ when you travel with Lorentz factor $\gamma$ ? (b) What is your measure of the time required for that travel? For the next questions, calculate travel times as multiples of $L / c$ to four significant figures. (c) Starting at junction $U$, what should the next three junctions be to minimize the travel time, and how much is that time? (d) What should the next two junctions be to minimize the travel time, and how
much is that time? (e) What next five junctions should you pass through to minimize the time and land on $E$ (Far Base), and how much is that time? (f) What is the total travel time?

Salamat Ali

Numerade Educator