Contention Task - Block and Bullet Speed After Impadt In Case A, a metal builet

Question

Contention Task - Block and Bullet Speed After Impadt In Case A, a metal builet penetrates a wooden block. In Case B, a rubber bullot with the same nitial speed and mass bounces off of an identical wooden block. Before After Case B Before After a) Will the speed of the wooden block after the colision be (0 greater in Case A, (i) greater in Case B, or (il) the same in both cases? Explain your reasoning b)In Case B, will the speed of the bullet after the colision be (i)greater than, (ii) less than, or(W) the same as the speed of the bullet just before the collision? Explain your reasoning.

Explanation / Answer

I assume you want an explanation for the reasoning mentioned. I will present a mathematical equivalent of the situation above and then try and explain the same.

Think of the bullet as of mass m and the block with a mass M. We will assume that the initial velocity of the bullet is u and the final velocity is v and the velocity of the wooden block to be V after the collision.

Plus, we know that in the absence of any external force or source of energy, the linear momentum and the total energy of the system at hand has to remain conserved.

Now, initial energy = 0.5*mu^2

initial linear momentum = mu

Case A: Here the block and bullet will have same final velocity, say, V after it penetrates it

Hence momentum = (M+m)v which has to be equal to mu

or v = mu / (m+M)

Case B: Here bullet will have a velocity backwards say, v1 and the block will have a velocity forward, V1

Hence mu = MV1 - mv1

or V1 = m(u + v1)/M

Compare this with the velocity in case A, you can easily see that the velocity will increase

Further, energy has to remain conserved.

That is, mu^2 = MV1^2 + mv1^2

for case A, we have: mu^2 = (m+M)v^2

now, as we have shown, V1 > v. So for the energy to be same as mu^2, v1 must be smaller than v.

Hence, speed of bullet must be smaller.

NOTE: Consider as though the bullet gives away its energy and momentum to the block. In case one, they both carry forward. In case B only block carries the momentum forward, hence mass moving forward is less so velocity must be higher.

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