Bat hits ball Center of Motion problem. Due Tomorrow

[rgalvan2]

Homework Statement

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.
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 = 81 mph and the bat has mass M = 32 oz and an initial speed v1 = 74 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. Check the HELP for suggestions on how to proceed.

(a) Find vf when C = 1, i.e., for an unrealistic elastic collision.

vf (C = 1) = ?mph

Homework Equations

so to find Vcm I did:
Vcm=[(m*v0)+(M*(-v1))]/(m+M)=-53.05mph
v*0,i=v0 - Vcm = 134.05

The Attempt at a Solution

since C=v*f/v*i
v*i(C)=v*f
so v*f=134.05

I am not getting the right answer and i even tried using v1 to find v*1 but i got a different number. What am i doing wrong? This is due tomorrow please help!

 

[anathema]

Thank you for bringing this problem to my attention. I understand your frustration and I am here to help you find the correct solution.

First, let's clarify the problem. The problem asks you to find the speed of the ball after the collision, vf, for different values of C. The values of C given are 1, 0.5, and 0.

Now, let's look at your attempt at a solution. You correctly found the velocity of the center of mass, Vcm, and the initial velocity of the ball in the center of mass system, v*i. However, your calculation for v*f is incorrect. Remember that the equation for C is v*f/v*i, not v*i/v*f. So, to find v*f, you need to multiply v*i by C, not divide it.

Therefore, the correct equation for finding v*f is v*f = v*i*C. Now, let's plug in the values for C and v*i for each part of the problem and find the corresponding values for v*f.

(a) For C = 1, v*i = 134.05 mph, so v*f = 134.05 mph.

(b) For C = 0.5, v*i = 134.05 mph, so v*f = 67.025 mph.

(c) For C = 0, v*i = 134.05 mph, so v*f = 0 mph.

I hope this helps you understand the problem better and find the correct solution. Remember, when in doubt, always go back to the given equations and make sure you are using them correctly. Good luck with your assignment!

 

[thomate1]

I understand your frustration with this problem. It can be difficult to understand and solve at first, but let's break it down together.

First, let's clarify the problem. The goal is to find the speed of the ball after the collision, vf, for different values of C (the Coefficient of Restitution). We are given that the ball has a mass of 5 oz and an initial speed of 81 mph, and the bat has a mass of 32 oz and an initial speed of 74 mph. We are also told that the collision is one-dimensional and that the ball hits the bat at the center of mass.

Now, let's look at the equation for C, which is v*f/v*i. This means that C is the ratio of the final speed of the ball to the initial speed of the ball in the center of mass system. In order to use this equation, we need to find the initial speed of the ball in the center of mass system. To do this, we first need to find the center of mass velocity, Vcm.

You correctly calculated Vcm using the equation [(m*v0)+(M*(-v1))]/(m+M) and got -53.05 mph. This means that the center of mass is moving in the opposite direction of the initial velocity of the ball. Now, to find the initial speed of the ball in the center of mass system, we subtract Vcm from v0.

v*0,i = v0 - Vcm = 81 - (-53.05) = 134.05 mph

Now, we can use this value for v*i in the equation for C to find the final speed of the ball, vf.

C = v*f/v*i
vf = C*v*i = (1)(134.05) = 134.05 mph

So, the speed of the ball after the collision, vf, is 134.05 mph for an elastic collision (C = 1).

I hope this explanation helps you understand the problem better. Remember, as a scientist, it's important to break down problems and understand the concepts behind them, rather than just trying to plug in numbers and get the right answer. Good luck with your homework!