Question

Two masses of 1 gram and 4 gram are moving with equal kinetic energy. The ratio of the magnitudes of their momenta is

• A. 4:1
• B. \( \sqrt{2}:1 \)
• C. 1:12
• D. 1:16

Hint:

In problems involving kinetic energy and momentum, remember that the ratio of the momenta depends on the square root of the ratio of the masses when the kinetic energies are equal.

Answer 1

The Correct Option is B

Step 1: Using the formula for kinetic energy.
Kinetic energy is given by: \[ K = \frac{1}{2} m v^2 \] Where: - \( m_1 = 1 \, \text{g} \), \( m_2 = 4 \, \text{g} \) - \( v_1 \) and \( v_2 \) are the velocities of the masses. Since the kinetic energies are equal: \[ \frac{1}{2} m_1 v_1^2 = \frac{1}{2} m_2 v_2^2 \] Simplifying, we get: \[ m_1 v_1^2 = m_2 v_2^2 \] Step 2: Solving for velocities.
We can solve for the velocities in terms of the masses: \[ v_1 = \sqrt{\frac{m_2}{m_1}} v_2 \] Substituting \( m_1 = 1 \, \text{g} \) and \( m_2 = 4 \, \text{g} \): \[ v_1 = \sqrt{4} v_2 = 2 v_2 \] Step 3: Calculating the momenta.
Momentum is given by \( p = mv \). Thus, the momenta of the two masses are: \[ p_1 = m_1 v_1 = 1 \times 2 v_2 = 2 v_2 \] \[ p_2 = m_2 v_2 = 4 \times v_2 = 4 v_2 \] The ratio of the momenta is: \[ \frac{p_1}{p_2} = \frac{2 v_2}{4 v_2} = \frac{1}{2} \] Thus, the ratio of the magnitudes of the momenta is \( \sqrt{2}:1 \), corresponding to option (B).