# Elastic collision in center of mass frame

1. Mar 24, 2009

### TwinGemini14

In this problem we will examine the collision between a baseball (the cowhide) and a bat (the ash). We assume a one-dimensional problem. That is, the bat hits the ball squarely, so that the ball reverses its direction after the collision. We also assume that the ball hits the bat at the center of mass of the bat. As you will learn later in the course, this means that we can ignore any effects due to the rotational motion of the bat.

The collision between the bat and ball is not an elastic collision. Instead it is characterized by a quantity known as the Coefficient of Restitution, which we shall denote in the problem by the symbol C. C is defined as follows. Suppose two objects collide. Let v*i and v*f be the speed of one of the objects in the Center of Mass (CM) system before and after the collision, respectively. Then

C = v*f/v*i
This means that C2 is the ratio of kinetic energy in the CM system after the collision to that before the collision. We know that for an elastic collision (see Lecture 15), the kinetic energy is conserved, so that C = 1 for perfectly elastic collisions. For a completely inelastic collision, C = 0.

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The following are three nearly identical problems that only differ in the value of C. In each case, the baseball has mass m = 5 oz and an initial speed v0 = 82 mph and the bat has mass M = 32 oz and an initial speed v1 = 78 mph. The basic problem is to find the speed of the ball after the collision, vf, for different values of C. You will probably find it useful to derive a general algebraic formula that relates vf to C and the various quantities given. Then you only have to plug into that formula the different values of C given below.
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A) Find vf when C = 1, i.e., for an unrealistic elastic collision.

1. So we first determine the velocity at the center of mass:

Vcm = ((m1v1+m2v2)/(m1+m2)) = (((5*-82)+(32*78))/(5+32)) = 56.38 m/s

2. Then we calculate the initial velocities in the center of mass reference frame:

v* = v-Vcm = 82-56.38 = 25.62 m/s

3. Next we find the final velocities in the center of mass reference frame:

v*f = -v*i = -25.62 m/s

4. Finally we calculate the final velocities in the lab reference frame:

v = v*+Vcm = -25.62+56.38 = 30.76 m/s

SO... vf = 30.76 m/s

2. Mar 24, 2009

### TwinGemini14

Never mind guys... I figured it out. Sorry about that.

3. Mar 29, 2009

### einstein18

Im stuck on the same problem, doing the same thing. What did you do wrong?