Question

A hard spherical ball “B” is at rest, and a second ball “A” is moving at a speed of 4.00 m/s, they collide linearly and elastically. If ball A rebounds with half its initial speed, what is the ratio of their masses (mA:mB) and what is the final velocity of ball B?

Answer 1

Let mass of ball A=mA and that of ball B=mB

Let final velocity of ball A=vA and that of ball B=vB

As collision is elastic, kinetic energy and momentum remains conserved.

From Moment conservation we get

mA*4 +0=mA*(-2) +mBvB...... (-2 because it rebounded and thus direction changed).

4mA+2mA=mBvB

6mA=mBvB........(say equation 1).

From kinetic energy conservation we get

1/2*mA*(4²)+0=1/2.(mA)*(2²)+1/2(mB)(vB)²

Cancel out 1/2

16mA-4mA=mB(vB)²

12mA=mB(vB)²...........(say equation 2).

From equation 1 we have mA=mBvB/6

12*(mBvB/6)=mB(vB)²

Cancel out mBvB we get

12/6=vB=2m/s i.e in the direction opposite to that of mA.

And from equation 1 we have mA/mB=vB/6

mA/mB=2/6=1/3.