Question

A light body and a heavy body have the same momentum .Which one has greater kinetic energy?
A. The light body
B. Both have equal K.E.
C. The heavy body
D. Data given is incomplete.

Answer 1

We know that both the bodies have the same momentum, we have to write the formula for momentum, which is in relation with the kinetic energy of a body, then compare the mass of both the heavy body and the lighter body to know which body has higher kinetic energy.

Complete answer:
As per the given question, we see that there is a heavy body and a light body now let their masses be ${{M}_{1}}$ and ${{M}_{2}}$ respectively.
Now we know that, ${{M}_{1}}$ > ${{M}_{2}}$
Given ${{P}_{1}}$ = ${{v}_{1}}$ ${{M}_{1}}$=${{v}_{2}}{{M}_{2}}$=${{P}_{2}}$
Since momentum is the same for the lighter as well as the heavier body then we can say that the lighter body moves with a greater velocity.
We know that momentum is the product of velocity and mass of a particle. It is usually a vector quantity and has both magnitude and direction; it is actually the quantity of motion of a moving body.
Now, we can mathematically relate momentum and kinetic energy in this process,
We know that $K.E=\dfrac{1}{2}m{{v}^{2}}$
Now we are multiplying and dividing the RHS with mass ‘m’ so,
We get,
$\Rightarrow K.E=\dfrac{1}{2}m{{v}^{2}}\times \dfrac{m}{m}=\dfrac{{{m}^{2}}{{v}^{2}}}{2m}=\dfrac{{{(mv)}^{2}}}{2m}$ ……….. Eq.1
Now we know that p = m v…..Eq.2.
So we are now substituting Eq2 in Eq1,
We get,
$K.E=\dfrac{{{p}^{2}}}{2m}$
We know,
K=$\dfrac{1}{2}m{{v}^{2}}$ or K=$\dfrac{{{p}^{2}}}{2m}$
Therefore,
$\dfrac{{{K}_{1}}}{{{K}_{2}}}=\dfrac{{{P}_{1}}^{2}{{M}_{2}}}{{{P}_{2}}^{2}{{M}_{1}}}=\dfrac{{{M}_{2}}}{{{M}_{1}}}$ ,…………. Eq.1
Since ${{M}_{1}}>{{M}_{2}}$, and both the bodies have the same momentum then,
We can write this as from Eq.1,
${{P}_{1}}^{2}{{M}_{1}}>{{P}_{1}}^{2}{{M}_{2}}$,
Then we can say that,
${{K}_{1}}$ $<$ ${{K}_{2}}$ So, ${{M}_{2}}$ has more kinetic energy.

So, the correct answer is “Option A”.

Additional Information:
The kinetic energy of an object is the energy that it possesses due to the virtue of its motion. It is defined as the work needed to accelerate a body of a given mass from rest to its stated velocity.
Linear momentum is defined as the product of a system's mass multiplied by its velocity.
The mass of a body is its inertia or resistance to dynamic motion. More precisely, it is a property of the body that determines the body's acceleration under the influence of a given force.
Inertia is a quantity expressing the body’s resistance to angular acceleration and is the sum of the product of the mass of each and every particle of the body with the square of its distance from the axis of rotation.

Note:
We can say/find the answer in two ways with respect to the velocity other with respect to Mass. We know that a lighter body always has more velocity than a heavier one. Only because we knew momentum is the same that’s why we could predict that Mass or velocity is the varying factor.