Two bodies having the same linear momentum have a ratio of kinetic energy as 16:9. Find the ratio of masses of these bodies.
Show Hint
When two bodies have the same linear momentum but different kinetic energies, the body with the greater kinetic energy must have a smaller mass. This relationship arises because kinetic energy depends on both mass and the square of velocity, whereas momentum depends linearly on both.
- \(\frac{9}{16}\)
- \(\frac{4}{3}\)
- \(\frac{3}{4}\)
- \(\frac{16}{9}\)
The Correct Option is A
Solution and Explanation
Let the masses of the two bodies be \( m_1 \) and \( m_2 \), and their velocities be \( v_1 \) and \( v_2 \) respectively.
Given:
\[
\frac{\text{KE}_1}{\text{KE}_2} = \frac{16}{9}
\]
\[
\text{Momentum}_1 = \text{Momentum}_2 \implies m_1 v_1 = m_2 v_2
\]
Step 1: Express Kinetic Energies
\[
\text{KE}_1 = \frac{1}{2} m_1 v_1^2
\]
\[
\text{KE}_2 = \frac{1}{2} m_2 v_2^2
\]
\[
\frac{\text{KE}_1}{\text{KE}_2} = \frac{\frac{1}{2} m_1 v_1^2}{\frac{1}{2} m_2 v_2^2} = \frac{m_1 v_1^2}{m_2 v_2^2} = \frac{16}{9}
\]
Step 2: Use Momentum Equality
\[
m_1 v_1 = m_2 v_2 \implies v_2 = \frac{m_1}{m_2} v_1
\]
Step 3: Substitute \( v_2 \) in the Kinetic Energy Ratio
\[
\frac{m_1 v_1^2}{m_2 \left( \frac{m_1}{m_2} v_1 \right)^2} = \frac{16}{9}
\]
\[
\frac{m_1 v_1^2}{m_2 \left( \frac{m_1^2}{m_2^2} v_1^2 \right)} = \frac{16}{9}
\]
\[
\frac{m_1}{m_2} \cdot \frac{m_2^2}{m_1^2} = \frac{16}{9}
\]
\[
\frac{m_2}{m_1} = \frac{16}{9}
\]
\[
\frac{m_1}{m_2} = \frac{9}{16}
\]
Step 4: Determine the Ratio of Masses
\[
\text{Ratio of masses} = m_1 : m_2 = 9 : 16
\]
Hence, the correct ratio of their masses is \( 9 : 16 \).
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Concepts Used:
Kinetic energy
Kinetic energy of an object is the measure of the work it does as a result of its motion. Kinetic energy is the type of energy that an object or particle has as a result of its movement. When an object is subjected to a net force, it accelerates and gains kinetic energy as a result. Kinetic energy is a property of a moving object or particle defined by both its mass and its velocity. Any combination of motions is possible, including translation (moving along a route from one spot to another), rotation around an axis, vibration, and any combination of motions.
