Question

The relation between momentum(P), mass(m), Kinetic energy(KE) and velocity(V) of the body is:
A. $PV=2KE$
B. $KE={\displaystyle \frac{P^2}{2m}}$
C. $P^2=2mKE$
D. All of the above.

Answer 1

- Hint: In this question, we should have some basic knowledge of the terms that are given in the question. We will use the definition and formulae of those terms to find out the relation between them.

Complete step-by-step solution -

Momentum, product of the mass of a particle and its velocity. Momentum is a vector quantity; i.e., it has both magnitude and direction.
Kinetic energy is the product of one-half of the mass of the body and the square of its velocity or speed. It is a scalar quantity.
We know that the momentum and the kinetic energy of a moving body are the body’s properties which is very much related to velocity.
We have, $KE={\displaystyle \frac{1}{2}}m{V}^2$ and $P=mV$
Now,
$$\begin{gathered} \Rightarrow KE={\displaystyle \frac{1}{2}}m{V}^2 \\ \Rightarrow 2KE=m{V}^2 \\ \Rightarrow 2KE=(mV)V \end{gathered}$$
Putting P = mV we get,
$\Rightarrow 2KE=PV$ ……….(i)
 Multiplying ‘m’ both sides, we get
$\Rightarrow 2mKE=PmV$, again put P=mV
$\Rightarrow 2mKE=P^2$ ………..(ii)
$\Rightarrow KE={\displaystyle \frac{P^2}{2m}}$ ……..(iii)
Hence, from equations (i), (ii) and (iii) we can say that options A, B and C are correct. Therefore option D – all of the above is correct.

Note: In this type of questions, first we have to write the definition and the formulae of the terms that are given in the question. Then we will find that either other term is occurring in the formulae of one term or not. If so , then we will put that term in the formula, and through we will get the relationship of these terms .