# Can a collision between a large and small mass result in final velocity greater than initial velocity?

Physics Asked by Jack Freeth on May 8, 2021

If you have some large mass with a velocity $$v$$, and some small stationary mass, is it possible for a collision to occur that results in the smaller mass having a velocity $$v_f> v$$, and perhaps the larger mass even stopping? i.e all momentum is transferred to the smaller mass? Or will the larger mass always retain some of the momentum?

The best way is to find out, using calculations, Suppose initially mass $$m$$ is stationary and mass $$M$$ is moving with speed $$u_1$$, then if after the elastic collision the velocities are $$v_2$$ and $$v_1$$ respectively then $$v_2-v_1=u_1$$ $$Mu_1=Mv_1+mv_2$$ $$v_1=frac{M+m}{M-m}u_1 text{and} v_2=frac{2M}{M-m}u_1$$

Is it possible for a collision to occur that results in the smaller mass having a velocity $$v_f>v$$?

$$v_2>u_1Rightarrow frac{2M}{M-m}u_1>u_1Rightarrow 2M>M-mRightarrow M>-m$$ which is true. So it's possible.

all momentum is transferred to the smaller mass? Or will the larger mass always retain some of the momenta?

Suppose the case, In which the $$v_1=0$$ so that $$M+m=0Rightarrow M=-m$$ which is not possible.

Correct answer by Young Kindaichi on May 8, 2021

It is possible under contrived conditions for all of the momentum to be transferred from a large mass object to a small mass object as you have described.

The issue is the energy. In such a collision the total kinetic energy of the system after the collision is greater than the total kinetic energy before the collision. So the collision must involve the conversion of some other form of energy into kinetic energy.

For example, suppose the smaller object consisted of a mass, a compressed spring, and a latch. If the larger mass hits the latch and releases the compressed spring which forcibly pushes the masses apart, then the condition you describe can be created.

Answered by Dale on May 8, 2021

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