Question

A ball of mass 10 g moving with 4 ms\(^{-1} \) collides with another ball of same mass at rest. If 0.2 is the coefficient of restitution of the collision, then the ratio of the velocity of first ball to that of the second ball is

• A. \( \frac{3}{2} \)
• B. \( \frac{2}{3} \)
• C. \( \frac{1}{4} \)
• D. \( \frac{1}{6} \)

Hint:

Apply the principle of conservation of linear momentum and the definition of the coefficient of restitution to set up a system of equations involving the final velocities of the colliding balls. Solve these equations simultaneously to find the final velocities and then calculate their ratio.

Answer 1

The Correct Option is B

Let the mass of the first ball be \( m_1 \) and its initial velocity be \( u_1 \).
Let the mass of the second ball be \( m_2 \) and its initial velocity be \( u_2 \).
Let the final velocity of the first ball be \( v_1 \) and the final velocity of the second ball be \( v_2 \).
Given: \( m_1 = 10 \) g \( u_1 = 4 \) ms\(^{-1} \) \( m_2 = 10 \) g \( u_2 = 0 \) ms\(^{-1} \) Coefficient of restitution \( e = 0.
2 \) According to the law of conservation of linear momentum: \( m_1 u_1 + m_2 u_2 = m_1 v_1 + m_2 v_2 \) \( 10(4) + 10(0) = 10 v_1 + 10 v_2 \) \( 40 = 10 v_1 + 10 v_2 \) \( 4 = v_1 + v_2 \) (Equation 1) The coefficient of restitution \( e \) is defined as the ratio of the relative velocity of separation to the relative velocity of approach: \( e = \frac{v_2 - v_1}{u_1 - u_2} \) \( 0.
2 = \frac{v_2 - v_1}{4 - 0} \) \( 0.
2 = \frac{v_2 - v_1}{4} \) \( 0.
8 = v_2 - v_1 \) (Equation 2) Now we have a system of two linear equations with two variables \( v_1 \) and \( v_2 \): 1) \( v_1 + v_2 = 4 \) 2) \( v_2 - v_1 = 0.
8 \) Add Equation 1 and Equation 2: \( (v_1 + v_2) + (v_2 - v_1) = 4 + 0.
8 \) \( 2 v_2 = 4.
8 \) \( v_2 = \frac{4.
8}{2} = 2.
4 \) ms\(^{-1} \) Substitute the value of \( v_2 \) into Equation 1: \( v_1 + 2.
4 = 4 \) \( v_1 = 4 - 2.
4 = 1.
6 \) ms\(^{-1} \) The ratio of the velocity of the first ball to that of the second ball is \( \frac{v_1}{v_2} \): \( \frac{v_1}{v_2} = \frac{1.
6}{2.
4} = \frac{16}{24} = \frac{2 \times 8}{3 \times 8} = \frac{2}{3} \)