Question

# A light and a heavy object have the same momentum. Find out the ratio of their kinetic energies. Which one has a larger kinetic energy?

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Solution

## Step 1. Given dataThe momentum of the light object and heavy object is the same.Step 2. Formula usedMomentum $=mv$Kinetic Energy$\left(K.E\right)$$=\frac{1}{2}m{v}^{2}$Where $m$ is the mass and $v$ is the velocity of the object.Step 3. Calculate the ratio of their kinetic energiesLet $m$ is the mass and $v$ is the velocity of the lighter object and $M$ is the mass and $V$ is the velocity of the heavy object.Since both have the same momentum$\therefore mv=MV\phantom{\rule{0ex}{0ex}}⇒\frac{M}{m}=\frac{v}{V}$Clearly $v>V$The kinetic energy of the lighter object will be be${\left(K.E\right)}_{1}=\frac{1}{2}m{v}^{2}$The kinetic energy of the heavier object will be${\left(K.E\right)}_{2}=\frac{1}{2}M{V}^{2}$$\therefore {\left(K.E\right)}_{1}:{\left(K.E\right)}_{2}\phantom{\rule{0ex}{0ex}}=\frac{1}{2}m{v}^{2}:\frac{1}{2}M{V}^{2}\phantom{\rule{0ex}{0ex}}=m{v}^{2}:\frac{mv}{V}{V}^{2}\left[\because \frac{m}{M}=\frac{V}{v}\right]\phantom{\rule{0ex}{0ex}}=v:V$So the ratio of the kinetic energy is $v:V$.Step 4. Find the larger kinetic energy$\because v>V$So ${\left(K.E\right)}_{1}>{\left(K.E\right)}_{2}$Hence, the ratio of the kinetic energies is $v:V$ and the kinetic energy of the lighter object is greater than the kinetic energy of the heavier object.

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