Question

The sphere A of mass \( m \) moving with a constant velocity hits another sphere B of mass \( 2m \) at rest. If the coefficient of restitution is 0.4, the ratio of the velocities of the spheres A and B after collision is

• A. 3:1
• B. 1:5
• C. 1:7
• D. 4:1

Hint:

Use both the coefficient of restitution and conservation of momentum to solve for velocities after collision.

Answer 1

The Correct Option is C

Let's solve the problem step by step: Let the mass of sphere A be $m_A = m$ and its initial velocity be $u_A = u$. Let the mass of sphere B be $m_B = 2m$ and its initial velocity be $u_B = 0$. The coefficient of restitution is $e = 0.4$. Let the final velocities of spheres A and B be $v_A$ and $v_B$, respectively. We have two equations: 1. Conservation of momentum: $m_A u_A + m_B u_B = m_A v_A + m_B v_B$ 2. Coefficient of restitution: $e = \frac{v_B - v_A}{u_A - u_B}$ Substituting the given values: 1. $m(u) + 2m(0) = m v_A + 2m v_B$ $mu = m v_A + 2m v_B$ $u = v_A + 2v_B$ (1) 2. $0.4 = \frac{v_B - v_A}{u - 0}$ $0.4u = v_B - v_A$ (2) From (1), we have $v_A = u - 2v_B$. Substituting this into (2): $0.4u = v_B - (u - 2v_B)$ $0.4u = v_B - u + 2v_B$ $0.4u = 3v_B - u$ $1.4u = 3v_B$ $v_B = \frac{1.4u}{3} = \frac{14u}{30} = \frac{7u}{15}$ Now, substitute $v_B$ back into the equation for $v_A$: $v_A = u - 2v_B = u - 2(\frac{7u}{15}) = u - \frac{14u}{15} = \frac{15u - 14u}{15} = \frac{u}{15}$ Now, find the ratio $v_A : v_B$: $\frac{v_A}{v_B} = \frac{\frac{u}{15}}{\frac{7u}{15}} = \frac{u}{15} \times \frac{15}{7u} = \frac{1}{7}$ Therefore, the ratio of the velocities of the spheres A and B after the collision is 1:7. Final Answer: The final answer is $\boxed{(3)}$