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(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.

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a)$p_{x}: m_{A} v_{A}=m_{A} v_{A}^{\prime} \cos \theta_{A}^{\prime}+m_{n} v_{n}^{\prime} \cos \theta_{n}^{\prime}$$p_{p}: \quad 0=m_{A} v_{A}^{\prime} \sin \theta_{A}^{\prime}-m_{n} v_{m}^{\prime} \sin \theta_{m}^{\prime}$b) $\approx46.9^{\circ}$$\approx 1.23 \mathrm{m} / \mathrm{s}$

Physics 101 Mechanics

Chapter 9

Linear Momentum

Motion Along a Straight Line

Kinetic Energy

Potential Energy

Energy Conservation

Moment, Impulse, and Collisions

Rutgers, The State University of New Jersey

University of Washington

Hope College

Lectures

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In physics, a conservative force is a force that is path-independent, meaning that the total work done along any path in the field is the same. In other words, the work is independent of the path taken. The only force considered in classical physics to be conservative is gravitation.

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In physics, the kinetic energy of an object is the energy which it possesses due to its motion. It is defined as the work needed to accelerate a body of a given mass from rest to its stated velocity. Having gained this energy during its acceleration, the body maintains this kinetic energy unless its speed changes. The same amount of work is done by the body in decelerating from its current speed to a state of rest. The kinetic energy of a rotating object is the sum of the kinetic energies of the object's parts.

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is it looks something like this. So initially you have Ah m a moving with speed V A. Uh, nmb is is arrest, but after the collision, you have m a moving at it. An angle data A from the x axis. Let's speed be a prime and M b moves a single day. Toby, uh, it's speed Vehbi Prime in the opposite in Why direction, but the same extraction So part And the equations are as follows. So, um, for the horizontal dimension, the X axis momentum in the X axis as conservatives follows m a v a initially, uh, because initially it's all horizontal motion eyes equal to m a p a prime, uh, co sign say, today plus and B B crime co signed a Toby. And the reason we're looking using the coastline components are is that the X axis is what? Well, you take the co sign of this and you get the X component of it. Similarly, when the white for the Y direction now you have no initial momentum on that is equal to the white component off A's final momentum. So m a V a prime signed data, uh minus minus because it is the opposite. Why direction M B V b prime signed data be okay, So these are two equations and, um pie Be re resolve them. So in part B, um, we want so we want the BB prime and data be prime. So that's right. Things out in terms of you be prints from equation from p of X. For this first equation party, we have N B B B prime co sign they should be is equal to M a times V F minus V a prime co sign data. Okay. And, ah, from the UAE equation, we have m B B B prime sign Data B is equal to m A V. A prime sign data that we're gonna call this one the 1st 1 equation one the 2nd 1 equation too. Ah. And so the trick here is to, um, divide up the two equations. So we you divide, um to buy warm. So what happens is that the, uh, M B V b prime cancel of Eby prime term cancels and you have signed data be over co sign. Hey, Toby, uh, is equal, which is equal to 10. Www um right. Senator Marco sinus tam. A sequel to, uh, the physical to M a times that. Whoops. Uh, yeah, many times. V a prime. Um, science data over and eight times V A minus V a prime. Ah, sign books. Co sign That s o the Emmys. Cancel. And so you have a V a prime scientific data over p A minus v a prime co sign fate, eh? Is he called to town? Saito beat. Therefore, they to be is just the art attention of all that sort of arc Tim v a prime final speed. It's 2.1 sign 30 because that's the angle. The ball a makes over via initial speed 2.8 minus 2.1 co sign 30. And this comes out to be 46.9 degrees. Right? And therefore, finally, we have become We can plug this back into equation one. Um oh, I am Or let's say equation too. Um, you have n b. That's equation too, by the way. Also, M b times V b prime, uh, and be be be prime. Uh, sign they'd be is equal are Well, now we know if a Toby, uh is 46.9. So and be the new crime sign 46.9 is equal to Ah, I am a V a primetime did m s 0.1 thio pointing one to kill gramps 0.12 kilograms. Uh, times, times v a snow times via which is RV a prime, which is 2.1 years per second times sign. Uh, 30 degrees. Uh, Therefore be be prime is 0.12 I'm not gonna write the units here. They're all consistent. 0.12 times you've done one times. I'm 30 over 300.14 programs. Uh, sign 46.9. What you get is maybe prime is 1.23 meters per second and that is it.

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