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 determine the ratio of the magnitudes of linear momentum for two bodies with equal kinetic energies, we need to understand the relationship between kinetic energy, mass, and momentum. 

Concepts Used:

• Kinetic energy of an object is given by the formula: \(KE = \frac{1}{2} mv^2\), where \(m\) is the mass and \(v\) is the velocity of the object.
• Linear momentum of an object is given by: \(p = mv\).

Given:

• Mass of body 1, \(m_1 = 4\) g
• Mass of body 2, \(m_2 = 25\) g
• Kinetic energies are equal: \(KE_1 = KE_2\)

Step-by-step Solution:

1. Write the expression for kinetic energy for each body:

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

2. Since the kinetic energies are equal, we equate the expressions:

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

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

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

3. From this, express the velocities in terms of each other:

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

4. Solve for the ratio of velocities:

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

5. The ratio of the magnitudes of linear momentum \((p_1/p_2)\) can be expressed as:

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

Substitute the values and expression for \(\frac{v_1}{v_2}\):

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

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

Conclusion:

The correct answer is 2:5. This matches the given correct answer in the options.

Answer 2

For objects with equal kinetic energies \(\left(\frac{p_1^2}{2m_1} = \frac{p_2^2}{2m_2}\right)\), we have:

\[\frac{p_1}{p_2} = \sqrt{\frac{m_1}{m_2}}\]

Substituting \(m_1 = 4 \, \text{g}\) and \(m_2 = 25 \, \text{g}\):

\[\frac{p_1}{p_2} = \sqrt{\frac{4}{25}} = \frac{2}{5}\]

Thus, the ratio of their momenta is 2 : 5.