Puck A (mass mA ) is moving on a frictionless, horizontal...

Puck $A$ (mass $m_{A} )$ is moving on a frictionless, horizontal air table in the $+x$ -direction with velocity $\vec{v}_{A 1}$ and makes an elastic, headon collision with puck $B\left(\text { mass } m_{B}\right),$ which is initially at rest. After the collision, both pucks are moving along the $x$ -axis. (a) Calculate the velocity of the center of mass of the two-puck system before the collision. (b) Consider a coordinate system whose origin is at the center of mass and moves with it. Is this an inertial reference frame? (c) What are the initial velocities $\vec{u}_{\Delta 1}$ and $\vec{u}_{B 1}$ of the two pucks in this center-of-mass reference frame? What is the total momentum in this frame? (d) Use conservation of momentum and energy, applied in the center-of-mass reference frame, to relate the final momentum of each puck to its initial momentum and thus the final velocity of each puck to its initial velocity. Your results should show that a one-dimensional, elastic collision has a very simple description in center-of-mass coordinates. (e) Let $m_{A}=0.400 \mathrm{kg}$ , $m_{B}=0.200 \mathrm{kg},$ and $v_{A 1}=6.00 \mathrm{m} / \mathrm{s}$ . Find the center-of-mass velocities $\vec{u}_{A 1}$ and $\vec{u}_{B 1},$ apply the simple result found in part (d), and transform back to velocities in a stationary frame to find the final velocities of the pucks. Does your result agree with Eqs. $(8.24)$ and $(8.25) ?$

Puck $A$ (mass $m_{A} )$ is moving on a frictionless, horizontal air table in the $+x$ -direction
with velocity $\vec{v}_{A 1}$ and makes an elastic, headon collision with puck $B\left(\text { mass } m_{B}\right),$ which is initially at rest. After the collision, both pucks are moving along the $x$ -axis. (a) Calculate the velocity of the center of mass of the two-puck system before the collision. (b) Consider a coordinate system whose origin is at the center of mass and moves with it. Is this an inertial reference frame? (c) What are the initial velocities $\vec{u}_{\Delta 1}$ and $\vec{u}_{B 1}$ of the two pucks in this center-of-mass reference frame? What is the total momentum in this frame? (d) Use conservation of momentum and energy, applied in the center-of-mass reference frame, to relate the final momentum of each puck to its initial momentum and thus the final velocity of each puck to its initial velocity. Your results should show that a one-dimensional, elastic collision has a very simple description in center-of-mass coordinates. (e) Let $m_{A}=0.400 \mathrm{kg}$ ,
$m_{B}=0.200 \mathrm{kg},$ and $v_{A 1}=6.00 \mathrm{m} / \mathrm{s}$ . Find the center-of-mass velocities $\vec{u}_{A 1}$ and $\vec{u}_{B 1},$ apply the simple result found in part (d),
and transform back to velocities in a stationary frame to find the final velocities of the pucks. Does your result agree with Eqs. $(8.24)$ and $(8.25) ?$

Key Concepts

Center of Mass

The center of mass is the weighted average position of all the mass in a system. For collision problems, computing the center-of-mass velocity simplifies analysis by providing a frame where the net momentum is zero, thus making the description of the collision more symmetric and easier to analyze.

Inertial Reference Frame

An inertial reference frame is one in which an object not subjected to external forces moves with a constant velocity. This concept is critical because the laws of conservation of momentum and energy hold in inertial frames, making analysis and transformations between frames straightforward.

Conservation of Momentum

This principle states that the total momentum of an isolated system remains constant in the absence of external forces. It is fundamental for collision analysis, as it allows the determination of unknown velocities after collision by equating the total momentum before and after the impact.

Conservation of Kinetic Energy (Elastic Collisions)

In an elastic collision, both momentum and kinetic energy are conserved. This double conservation provides additional constraints needed to solve for the final velocities, and it simplifies the problem when transitions are made to the center-of-mass frame.

Galilean Transformation

Galilean transformations are used to switch between different inertial reference frames moving at constant velocities relative to one another. They are crucial in transforming velocities from the laboratory frame to the center-of-mass frame and back, ensuring that the collision analysis remains consistent across different frames.

Elastic Collision Analysis in the Center-of-Mass Frame

Analyzing an elastic collision in the center-of-mass frame reduces the complexity of the problem because the net momentum in this frame is zero. This makes the exchange of momentum and kinetic energy between the colliding bodies symmetric, simplifying the derivation of the final velocities.

Transcript

00:01 Hello stan, in this question we have given that two hockey players, okay? one is along vaxes.

00:09 One is along, sorry, one is along vaxes, it is two and one is along x -axis but making an angle 30 degree with the origin like that.

00:21 Okay, this our hockey player 1 and this our hockey player 2.

00:25 Okay, this angle is 30 degree.

00:28 This player has velocity 20 meters per second, 20 meter per second.

00:32 Okay and this player have velocity 15 meter per second in by direction.

00:37 Okay, let's say this is u2, this over u1.

00:42 Now there is collide with each other at this point and it is given that if after collision both players one and two get stuck that is they stuck with each other and starts moving the same velocity.

00:55 Now it means condition when they stuck each other, his momentum is conserved.

01:00 So this whole first question that it is is momentum.

01:03 Is conserved for this condition and during collision during collision net force on the system net force on the system is zero so momentum is always conserved so momentum is conserved momentum is conserved okay if we see these two hockey players like that this is our first player and this our second player it is moving in this direction and second ball or second player is coming in this direction now during collision that is along the line of impact both these players will experience impulsive normal force and d t and d t so net force is zero therefore momentum is concerned and this collision is inelastic inelastic because in case of in elastic after collision both balls or mass starts moving this other starts moving simply we can say starts moving with the same velocity with the same velocity okay same velocity that is if two masses collide with each other if up to collision they move with the same velocities then it will be called elastic collision okay v and v if these two balls combined together after collision and starts moving with the same velocity then if this is the velocity of the system then ball one also have velocity v and second ball also have velocity v so either in case of elastic collision after collision, velocity of both mass are same.

02:51 So collision is over inelastic.

02:53 And we have to find out the velocities of the mass after collision.

02:58 So momental is consulose.

02:59 So first momentum is conserved in x direction.

03:03 Okay? in x direction, let's see.

03:06 This velocity component will have a component, sorry, this velocity 20 meter per second.

03:10 That is player one...