To solve this problem, we need to apply the principles of conservation of linear momentum and the concept of coefficient of restitution after the collision between two spheres.
Principles and Explanation:
- Conservation of Linear Momentum: In the absence of external forces, the total linear momentum before and after the collision is conserved.
- Coefficient of Restitution (e): It is defined as the relative velocity of separation divided by the relative velocity of approach. Mathematically, \(e = \frac{v_2 - v_1}{u - 0}\), where \(v_2\) and \(v_1\) are the velocities of the two bodies after the collision.
Step-by-step Solution:
- Consider two spheres, sphere A and sphere B, both having mass \(m\). Sphere A is moving with velocity \(u\), and sphere B is stationary.
- From the conservation of linear momentum: \(mu = mv_1 + mv_2\)
Simplifying, we have: \(u = v_1 + v_2\) (Equation 1) - The coefficient of restitution is given by: \(e = \frac{v_2 - v_1}{u}\)
Simplifying this, we get: \(v_2 - v_1 = eu\) (Equation 2) - Adding Equation 1 and Equation 2: \(u + v_2 - v_1 = v_1 + v_2 + eu\)
Simplifying gives: \(2v_2 = u(1 + e)\)
Therefore: \(v_2 = \frac{u(1 + e)}{2}\) - Substituting the value of \(v_2\) in Equation 1: \(u = v_1 + \frac{u(1 + e)}{2}\)
Solving for \(v_1\): \(v_1 = \frac{u(1 - e)}{2}\) - Now, the ratio of velocities after collision is: \(\frac{v_1}{v_2} = \frac{\frac{u(1 - e)}{2}}{\frac{u(1 + e)}{2}}\)
Simplifying, we get: \(\frac{v_1}{v_2} = \frac{1 - e}{1 + e}\)
Thus, the ratio of the velocities \(v_1/v_2\) after the collision is given by \(\frac{1-e}{1+e}\). The correct option is:
\(\frac{1-e}{1+e}\)