Question

A light and a heavy object have the same momentum. Which one of the following statements is correct regarding the kinetic energy of these two objects?

• A. $(K.E{)}_l>(K.E{)}_h$
• B. $(K.E{)}_l=(K.E{)}_h$
• C. $(K.E{)}_h=1.5\times (K.E{)}_l$
• D. $(K.E{)}_h=2\times (K.E{)}_l$

Answer 1

The correct option is A

$(K.E{)}_l>(K.E{)}_h$

Momentum $=P=mv$
where $m=$ mass of the object and $v=$ velocity of the object

Kinetic Energy, KE $=\frac{1}{2}m{v}^2$
$=\frac{1}{2}\frac{m^2{v}^2}{m}(bymultiplyinganddividingbym$)
$=\frac{1}{2}\frac{{\left(mv\right)}^2}{m}=\frac{p^2}{2m}$
Hence, K.E $=\frac{p^2}{2m}$

Let kinetic energy of lighter and heavier objects be $K.E_l$ and $K.E_h$, respectively.
$\frac{K.E_l}{K.E_h}=\frac{\frac{P_l^2}{2m_l}}{\frac{P_h^2}{2m_h}}=\frac{m_h}{m_l}(Since{P}_l=P_h)$
where $P_l$ and $P_h$ are momentum and $m_l$ and $m_h$ are the mass of lighter and the heavier object, respectively.
Now, $m_h>m_l$
$\therefore K.E_l>K.E_h$
Hence, kinetic energy of lighter object is greater than heavier object.