Question

Two bodies having the same linear momentum have a ratio of kinetic energy as 16:9. Find the ratio of masses of these bodies.

• A. \(\frac{9}{16}\)
• B. \(\frac{4}{3}\)
• C. \(\frac{3}{4}\)
• D. \(\frac{16}{9}\)

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.

Answer 1

The Correct Option is A

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 \( 9 : 16 \).