Question

When a light and a heavy body have equal K.E., then which one has a greater momentum?
(A) Light body
(B) Heavy body
(C ) Both have equal momentum
(D) Uncertain

Answer 1

Hint: The momentum of a body depends upon its mass and velocity. Hence the body which has higher mass or velocity will have greater value of momentum. The momentum has the same direction as velocity since it directly depends on velocity. Momentum can also be described as a type of speed or force of a moving object.

Complete answer:
Let the mass of the lighter body be ${{m}_{1}}$ and that of the heavier body is ${{m}_{2}}$. Given that the kinetic energy of both the bodies are equal.
That is,
$\dfrac{1}{2}{{m}_{1}}v_{1}^{2}=\dfrac{1}{2}{{m}_{2}}v_{2}^{2}$
where, ${{m}_{2}}\rangle {{m}_{1}}$.
Rearranging the equation we get,
$\dfrac{v_{1}^{2}}{v_{2}^{2}}=\dfrac{{{m}_{2}}}{{{m}_{1}}}$
$\Rightarrow \dfrac{{{v}_{1}}}{{{v}_{2}}}=\sqrt{\dfrac{{{m}_{2}}}{{{m}_{1}}}}$
The momentum of the light body, ${{p}_{1}}={{m}_{1}}{{v}_{1}}$
The momentum of the heavy body, ${{p}_{2}}={{m}_{2}}{{v}_{2}}$
Hence,
$\begin{align}
  & \dfrac{{{p}_{1}}}{{{p}_{2}}}=\dfrac{{{m}_{1}}{{v}_{1}}}{{{m}_{2}}{{v}_{2}}} \\
 & \Rightarrow \dfrac{{{p}_{1}}}{{{p}_{2}}}=\dfrac{{{m}_{1}}}{{{m}_{2}}}\sqrt{\dfrac{{{m}_{2}}}{{{m}_{1}}}} \\
\end{align}$
$\begin{align}
  & \Rightarrow \dfrac{{{p}_{1}}}{{{p}_{2}}}=\sqrt{\dfrac{m_{1}^{2}}{m_{2}^{2}}\times \dfrac{{{m}_{2}}}{{{m}_{1}}}} \\
 & \therefore \dfrac{{{p}_{1}}}{{{p}_{2}}}=\sqrt{\dfrac{{{m}_{1}}}{{{m}_{2}}}} \\
\end{align}$
This relation shows that momentum is directly proportional to its mass.
Hence, the heavy body has greater momentum.
The momentum of a body depends upon its mass and velocity. Hence the body which has higher mass or velocity will have greater value of momentum. Thus momentum of a particle may be explained as the product of the mass of the body and its velocity.
According to Newton’s second law of motion force is proportional to the acceleration. Or otherwise, change in momentum is equal to impulse. Where impulse is the product of force and time. For a rigid body the total momentum of the body will be the sum of individual moments of particles. Since, velocity is a vector quantity the momentum is also a vector quantity. That is, which have both magnitude as well as direction. The momentum has the same direction as velocity since it directly depends on velocity. Momentum can also be described as a type of speed or force of a moving object. Whereas, the inertia is the tendency of an object to move on. Thus all the moving objects will have a momentum. The momentum of a body depends upon its mass and velocity. Hence the body which has higher mass or velocity will have greater value of momentum. Thus momentum of a particle may be explained as the product of its mass and velocity.

Note:
The momentum of a body depends upon its mass and velocity. Hence the body which has higher mass or velocity will have greater value of momentum. Thus momentum of a particle may be explained as the product of the mass of the body and its velocity. For a rigid body the total momentum of the body will be the sum of individual moments of particles.