How will the momentum of a body change if we double its kinetic energy? Prove by using equations.

Hello!

For a body having the mass `m` and the speed `V`, its kinetic energy `E` is equal to

`(mV^2)/2,`

and its momentum (impulse) `J` is equal to `mV.`

Doubling the kinetic energy of a body probably is achieved by changing its speed. If originally a body had the speed `V_0,` then its kinetic energy was `E_0=(m*V_0^2)/2,`

and its impulse was `J_0=m*V_0.`

And if we change its speed to `V_1` in such a way that the energy doubles,

`E_1=(m*V_1^2)/2=2E_0=mV_0^2,`

then `V_1=sqrt(2)*V_0.` And the new momentum will be

`J_1=m*V_1=sqrt(2)*m*V_0=sqrt(2)*J_0.`

The answer: if the kinetic energy of a body doubles, then its impulse is multiplied by `sqrt(2).` Note that `sqrt(2)` is approximately 1.4.

P.S. If we double the kinetic energy by changing the mass, then it must be doubled also and the momentum will be also doubled.