Question

Two bodies of mass 4 g and 25 g are moving with equal kinetic energies. The ratio of the magnitude of their linear momentum is:

• A. 3:5
• B. 2:5
• C. 5:4
• D. 4:5

Answer 1

The Correct Option is B

To find the ratio of the magnitudes of linear momentum for two bodies with identical kinetic energies, one must first understand the relationships connecting kinetic energy, mass, and momentum.

Concepts Used:

• The kinetic energy of an object is calculated using the formula: \(KE = \frac{1}{2} mv^2\), where \(m\) represents mass and \(v\) denotes velocity.
• Linear momentum of an object is defined as: \(p = mv\).

Given:

• Mass of body 1: \(m_1 = 4\) g
• Mass of body 2: \(m_2 = 25\) g
• Equal kinetic energies: \(KE_1 = KE_2\)

Step-by-step Solution:

1. Formulate the kinetic energy expression for each body:

• Body 1: \(KE_1 = \frac{1}{2} m_1 v_1^2\)
• Body 2: \(KE_2 = \frac{1}{2} m_2 v_2^2\)

2. Equate the expressions due to equal kinetic energies:

\(\frac{1}{2} m_1 v_1^2 = \frac{1}{2} m_2 v_2^2\)

Remove the \(\frac{1}{2}\) factor and rearrange the equation:

\(m_1 v_1^2 = m_2 v_2^2\)

3. Express the velocities relative to each other:

\(v_1^2 = \frac{m_2}{m_1} v_2^2\)

4. Determine the ratio of velocities:

\(\frac{v_1}{v_2} = \sqrt{\frac{m_2}{m_1}} = \sqrt{\frac{25}{4}} = \frac{5}{2}\)

5. Express the ratio of the magnitudes of linear momentum (\((p_1/p_2)\)) as follows:

\(\frac{p_1}{p_2} = \frac{m_1 v_1}{m_2 v_2}\)

Substitute the given values and the derived velocity ratio:

\(\frac{p_1}{p_2} = \frac{4 \cdot \frac{5}{2}v_2}{25 \cdot v_2} = \frac{20}{50} = \frac{2}{5}\)

The resulting ratio of their linear momentum is \(2:5\).

Conclusion:

The calculated ratio is 2:5, which aligns with the provided correct answer.