Game of Pool In a game of pool, the cue ball strikes another...

Game of Pool In a game of pool, the cue ball strikes another ball of the same mass and initially at rest. After the collision, the cue ball moves at $3.50 \mathrm{~m} / \mathrm{s}$ along a line making an angle of $22.0^{\circ}$ with its original direction of motion, and the second ball has a speed of $2.00 \mathrm{~m} / \mathrm{s}$. Find (a) the angle between the direction of motion of the second ball and the original direction of motion of the cue ball and (b) the original speed of the cue ball. (c) Is kinetic energy (of the centers of mass, don't consider the rotation) conserved?

Key Concepts

Conservation of Linear Momentum

This principle states that in an isolated system, the total momentum remains constant throughout any collision. In collision problems, especially in two dimensions, momentum must be conserved along each independent direction, which allows for the setting up of equations to solve for unknown speeds or angles after the collision.

Vector Decomposition

When dealing with collisions occurring at angles, it is necessary to break vector quantities, such as velocity and momentum, into perpendicular components (usually horizontal and vertical). This allows one to apply conservation laws independently along each axis and accurately account for the directional nature of the motion.

Conservation of Kinetic Energy

This concept is applicable in elastic collisions where the total kinetic energy of the system remains unchanged before and after the collision. Checking for conservation of kinetic energy in a collision problem helps determine whether the collision is elastic or inelastic, providing insight into the energy dynamics of the impact.

Elastic Collision

An elastic collision is one in which both momentum and kinetic energy are conserved. In such collisions, especially in cases involving objects of equal mass, certain predictable outcomes occur in terms of speeds and deflection angles, simplifying the analysis of the system’s behavior after the collision.

Transcript

00:01 Hello.

00:02 So for this problem, our cue ball is striking another ball, and then the cue ball moves off with a speed of 3 .5 meters per second at an angle of 22 degrees with respect to its original motion.

00:15 The other ball moves off with a speed of 2 meters per second at some other angle.

00:21 And so for this, we're going to write two conservation and momentum equations, and then we're going to use those to solve the rest of the other angle.

00:31 Things so the first conservation momentum is in the x direction where the mass of the cue ball times its speed original speed we'll just call that v it's going to equal the cue ball times well these are all they all have the same mass so we'll just call them and the q q ball times it's times the of that angle because it's, or sorry, the cosine times a cosine of that angle plus the mass of the other ball times its speed times the cosine of its angle, which we'll call fee.

01:24 And then in the y direction, we're going to have no momentum initially.

01:33 And then we have mass of the cube ball times vc sine theta plus mass of cube of ball b times vb sine sine of fee so we can use this second equation to find out what fee is so we can set these equal to each other the masses are going to cancel out and so we can solve for this angle here by dividing both sides by bb and then taking the inverse sign of both sides so we get the fee is equal to inverse sign of vc sine theta over vb.

02:27 If you plug your values in for that, you'll get an angle of 40 .96 degrees with respect to the horizontal.

02:40 So that's down this way.

02:45 Now part b, we need to find the original speed.

02:48 And so for this, we can just use the top equation.

02:51 So mv is equal to mvc cosine theta plus mvb cosine phi, which we now know.

03:03 Now the masses are all going to cancel because they're all the same.

03:07 And then we have our equation that is already solved for the thing that we want...