Question

Two masses of 1 g and 9 g are moving with equal kinetic energies. The ratio of the magnitudes of their respective linear momenta is

• A. 1 : 9
• B. 9 : 1
• C. 1 : 3
• D. 3 : 1

Answer 1

The Correct Option is C

To solve this problem, we need to understand the relationship between kinetic energy and linear momentum. We are given that two masses have equal kinetic energies, and we need to find the ratio of their momenta.

Step-by-Step Solution:

1. The kinetic energy E_k of an object with mass m and velocity v is given by:
   E_k = \frac{1}{2} m v^2
2. The momentum p of an object is given by:
   p = m v
3. Since the kinetic energies of the two masses are equal, we can equate their kinetic energy expressions:
   For the 1 g mass: E_k = \frac{1}{2} \times 1 \times v_1^2
   For the 9 g mass: E_k = \frac{1}{2} \times 9 \times v_2^2
   Equating them:
   \frac{1}{2} \times 1 \times v_1^2 = \frac{1}{2} \times 9 \times v_2^2
4. Cancel the common terms and solve for v_1^2 and v_2^2:
   v_1^2 = 9 \times v_2^2
   v_1 = 3 \times v_2 (since v \geq 0)
5. Substitute back to find the ratio of momenta:
   For the 1 g mass, p_1 = 1 \times v_1 = 1 \times 3 \times v_2 = 3 v_2
   For the 9 g mass, p_2 = 9 \times v_2
6. The ratio of their momenta is:
   \frac{p_1}{p_2} = \frac{3 v_2}{9 v_2} = \frac{1}{3}
   Therefore, the ratio of the magnitudes of their respective linear momenta is 1:3.

Thus, the correct answer is 1 : 3.