Question

Assertion: If a light body and a heady possess the same momentum, the lighter body will possess more kinetic energy.
Reason: The kinetic energy of a body varies as the square of its velocity.
A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion.
B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion.
C. Assertion is correct but Reason is incorrect.
D. Both Assertion and Reason are incorrect.

Answer 1

Hint: Use the formula for kinetic energy of an object. This formula gives the relation between kinetic energy of the object, linear momentum of the object and mass of the object. Using this formula, derive the relation between the kinetic energy and masses of the two bodies as the linear momentum is the same for two bodies. Use the formula for kinetic energy of an object in terms of its velocity. Check if Assertion and Reason are correct and Reason gives the explanation of Assertion.

Formula used:
The kinetic energy \[K\] of an object is given by
\[K = \dfrac{{{P^2}}}{{2m}}\] …… (1)
Here, \[P\] is linear momentum of an object and \[m\] is mass of the object.

Complete step by step answer:
We have given that a light body and a heavy body has the same linear momentum and the kinetic energy of the light body is more than the heavy body.From the equation (1), we can conclude that the kinetic energy of an object is directly proportional to the square of linear momentum of the object and inversely proportional to mass of the object.
\[K \propto \dfrac{{{P^2}}}{m}\]

Since the linear momentum of the light body and heavy body is the same, the kinetic energy of the light and heavy body is inversely proportional to mass of the light and heavy bodies.
\[K \propto \dfrac{1}{m}\]
The mass of the light body is less than the mass of the heavy body. Hence, the kinetic energy of the light body is more than the kinetic energy of the heavy body.Hence, the statement given in Assertion is correct.

The kinetic energy of an object is given by
\[K = \dfrac{1}{2}m{v^2}\]
From the above equation, we can conclude that the kinetic energy of an object is proportional to square of velocity of the object.Hence, the Reason is also correct.But we can see that the Reason is not the correct explanation of Assertion.

Hence, the correct option is B.

Note:The students may think that since the linear momenta of the two bodies are the same, the change in kinetic energies of the two bodies is due to change in velocity of the body. Hence, the Reason correctly explains the Assertion. But the students should keep in mind that according to equation (1), the change in kinetic energies of two bodies is due to change in masses of the two bodies. Hence, the Reason does not correctly explain the Reason.