Question

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

Answer 1

Hint: The two bodies have different masses but the same kinetic energy. The kinetic energy of a body is related to both momentum and mass of the body. By comparing the masses with momentums we can get the required answer.

Detailed step by step solution:
The kinetic energy is the energy possessed by a body due to its state of motion. The expression for kinetic energy is given as
\[K = \dfrac{1}{2}m{v^2} = \dfrac{{{p^2}}}{{2m}}\]
Here m represents the mass of the body while v represents the velocity with which the body is moving and p is used to signify the momentum of the body. The kinetic energy can be expressed in terms of momentum by using the following expression where velocity can be represented in terms of momentum and mass.
$
  p = mv \\
  v = \dfrac{p}{m} \\
$
Now we have a light body and a heavy body which have the same kinetic energy. If ${m_1}$ is the mass of the lighter body and ${m_2}$ is the mass of the heavier body then we have
${m_1} < {m_2}$
If ${K_1}$ is the kinetic energy of the lighter body and ${K_2}$ is the kinetic energy of the heavier body then we have
${K_1} = {K_2}$
Now we can write the kinetic energies as follows using the expression for kinetic energy
$
  \dfrac{{p_1^2}}{{2{m_1}}} = \dfrac{{p_2^2}}{{2{m_2}}} \\
   \Rightarrow \dfrac{{{p_1}}}{{{p_2}}} = \sqrt {\dfrac{{{m_1}}}{{{m_2}}}} \\
  {m_1} < {m_2} \\
  \therefore {p_1} < {p_2} \\
$
This expression means that the lighter body has momentum less than the momentum of the heavier body. Hence the correct answer is option A.

Note: A body can have two types of energies, namely, kinetic energy and potential energy. Potential energy is energy due to the position of the body. Kinetic energy is possessed by a body only when it is motion. A body at rest has zero kinetic energy but non-zero potential energy.