Question

Two bodies of masses m and 2m have the same momentum. Their respective kinetic energies ${K_1}$ and ${K_2}$ are in the ratio-
A. $1:2$
B. $2:1$
C. $1:\sqrt 2 $
D. $1:4$

Answer 1

Hint: We can solve this problem with the concept of kinetic energy and momentum. Kinetic energy and momentum are related to each other. In the given question, we know that momentum of the two bodies of mass $m$ and $2m$ is the same.

Complete step by step answer:
Momentum is defined as the rate of change of the object’s position with respect to a frame of reference and time.The SI unit for momentum is $kg.m/s$. Momentum is directly proportional to the object’s mass and also its velocity. Thus the greater an object’s mass, the greatest its momentum. Momentum is a vector having the same direction as the velocity.
Formula for the momentum is $p = \sqrt {2mK} $ where $K$ is the kinetic energy.
To find the ratio of kinetic energy, we must know its concept-
Kinetic energy is a form of energy that an object or a particle has by reason of its motion.
Also we know the formula of kinetic energy i.e. $\dfrac{1}{2}m{V^2}$ where $V$ is the velocity and $m$ is the mass of the body.
From the formula of momentum, we get the following relationship of kinetic energy and momentum i.e. $K = \dfrac{{{p^2}}}{{2m}}$
We know that mass of the two bodied are $m,2m$ and the momentum is same for both then,
Kinetic energy for body $m$is ${K_1} = \dfrac{{{p^2}}}{{2m}}$
Kinetic energy for body $2m$ is ${K_2} = \dfrac{{{p^2}}}{{2 \times 2m}}$
Then the ratio of both the kinetic energy; $\dfrac{{{K_1}}}{{{K_2}}} = \dfrac{{{p^2}}}{{2m}}/\dfrac{{{p^2}}}{{2 \times 2m}}$
$\dfrac{{{K_1}}}{{{K_2}}} = \dfrac{{{p^2}}}{{2m}} \times \dfrac{{2 \times 2m}}{{{p^2}}} = \dfrac{2}{1}$
${K_1}:{K_2} = 2:1$
Therefore for the given question option B is correct.

Note:
While solving this question, we should also know that the formula of kinetic energy i.e. $\dfrac{1}{2}m{V^2}$ but we use this formula when the velocity is same in the question and if it is given that the momentum is same then we use $K = \dfrac{{{p^2}}}{{2m}}$.