Question

Give a reason for the following:
Can a body have momentum without having energy? explain.

Answer 1

Hint: Momentum is defined to be the mass of an object multiplied by the velocity of the object. So to define momentum we need to have its mass and its velocity as per its standard definition. Energy on the other hand is defined as the ability of a body to do work.

Complete step by step answer:
Writing mathematically, \[\overrightarrow{p}=m\overrightarrow{v}\]
From this, it is clear that momentum is a vector quantity and for a body to possess momentum it must be in motion. Now let us talk about energy. Energy can be of a different kind but we are concerned about mechanical energy and it is divided into potential energy and kinetic energy.
Kinetic energy is given by \[{{K}_{E}}=\dfrac{m{{v}^{2}}}{2}\] and potential energy is given by \[U=mgh\]. From the two formulas, it is clear that kinetic energy is the energy associated with motion and potential energy is the energy due to the position of the body. Now coming to the question can a body have momentum without having energy? Let us find a relationship between kinetic energy and momentum.
\[\begin{align}
& \overrightarrow{p}=m\overrightarrow{v} \\
&\Rightarrow {{K}_{E}}=\dfrac{m{{v}^{2}}}{2} \\
&\Rightarrow m{{K}_{E}}=\dfrac{{{m}^{2}}{{v}^{2}}}{2} \\
&\therefore {{K}_{E}}=\dfrac{{{p}^{2}}}{2m} \\
\end{align}\]

So, if the Kinetic energy is zero, momentum will be zero. Hence, a body cannot have momentum without having energy.

Note:We cannot associate momentum with potential energy because the prerequisite for a body to have potential energy is it must be at rest and for the body to have momentum it must be in motion, so when we talk about the momentum we can continue talking about kinetic energy and vice versa.