Question

A heavy and a light object have the same momentum. Which one has larger kinetic energy?
A) Heavier object
B) Lighter object
C) Both are equal
D) Cannot be determined

Answer 1

Hint: The momentum of an object is a physical quantity that refers to the quantity of motion of an object as is kinetic energy. Momentum is the product of the mass and velocity of an object.

Formula used: In this solution, we will use the following formula
Momentum of an object $ P = mv $ where $ m $ is the mass of the object and $ v $ is its velocity
Kinetic energy of an object: $ K = \dfrac{1}{2}m{v^2} $ .

Complete step by step answer
We’ve been given that a heavy and a light object have the same momentum. So, it implies that
 $ {P_{heavy}} = {P_{light}} $
So,
 $ {m_{heavy}}{v_{heavy}} = {m_{light}}{v_{light}} $
Now since $ {m_{heavy}} > {m_{light}} $ , for the momentum of the two objects to be equal, we must have
 $ {v_{light}} > {v_{heavy}} $
Now that we know this, let us further calculate the kinetic energy of the two objects. Using the formula for kinetic energy, we can write
 $ {K_{heavy}} = \dfrac{1}{2}mv_{heavy}^2\,{\text{and}} $
 $ {K_{light}} = \dfrac{1}{2}mv_{light}^2 $
Substituting $ P = mv $ for the respective masses, we can write
 $ {K_{heavy}} = \dfrac{1}{2}{P_{heavy}}{v_{heavy}}\,{\text{and}} $
 $ {K_{light}} = \dfrac{1}{2}{P_{light}}{v_{light}} $
Now both the particles have the same momentum i.e. $ {P_{heavy}} = {P_{light}} $ but the lighter mass has a higher velocity i.e. $ {v_{light}} > {v_{heavy}} $ so it will also have kinetic energy as it is the product of momentum and velocity of the object.
So the lighter mass will have higher kinetic energy. Hence the correct choice is option (B).

Note
It is common to expect the kinetic energy of the object to be the same if the momentum of the two objects is the same. But in our case, the masses, as well as the velocity of the two objects, are different. Also, the dependence of momentum and kinetic energy of velocity is different so we must use their formula to check which object will have higher kinetic energy.