Question

The momentum P and kinetic energy E of a body of mass m are related as:
(A) $P = \sqrt {2mE} $
(B) $P = \dfrac{1}{2}mE$
(C) $P = \dfrac{{2m}}{E}$
(D) $P = 2mE$

Answer 1

Hint
Kinetic energy of a body depends directly on the square of its linear momentum. The linear momentum is directly dependent on the mass and velocity of a body.
$\Rightarrow E = \dfrac{1}{2}m{v^2}$, where E is the kinetic energy, m is the mass of the body and v is the velocity. The SI unit of energy is Joules (J).

Complete step by step answer
Kinetic energy is the energy possessed by any particle or body when it accelerates, whereas the linear momentum is an object's mass in motion. Due to the dependency of these quantities on the motion, these are also dependent on each other. In this question, we are asked to find that relation.
We know that the kinetic energy of a body is given as:
$\Rightarrow E = \dfrac{1}{2}m{v^2}$
To manipulate this equation to get our desired form, we multiply both sides by m:
$\Rightarrow E.m = \dfrac{1}{2}{m^2}{v^2} = \dfrac{1}{2}{\left( {mv} \right)^2}$ [Eq. 1]
Now, we know that the momentum is given us:
$\Rightarrow P = mv$
Substituting this value in the Eq. 1 gives us:
$\Rightarrow E.m = \dfrac{1}{2}{P^2}$
$\Rightarrow 2mE = {P^2}$
This gives us the final relation as:
$\Rightarrow P = \sqrt {2mE} $
Hence, the correct answer is option (A).

Note
Kinetic energy of a body is responsible for any amount of work that it does. Almost all the objects in motion around us possess some kinetic energy. For example, a moving bicycle, or a swimmer in a swimming pool. While linear momentum tells about the inertia of an object in motion, and how much force will be required to bring it to a stop in a given amount of time.