Mechanical Properties Of Solids

We are tasked with finding the coefficient of restitution ( ) when a ball strikes a horizontal floor at an angle of and loses of its kinetic energy during the collision.
Step 1: Understand the problem The ball strikes the floor at an angle of . of its kinetic energy is lost during the collision. The coefficient of restitution is defined as the ratio of the relative velocity after the collision to the relative velocity before the collision, along the line of impact.
Step 2: Analyze the collision When the ball strikes the floor at , its velocity can be resolved into two components:
1. Horizontal component: ,
2. Vertical component: .
During the collision:
The horizontal component remains unchanged because there is no force acting horizontally. The vertical component changes due to the collision. If is the coefficient of restitution, the vertical velocity after the collision is . Step 3: Kinetic energy before and after the collision The initial kinetic energy ( ) is: The final kinetic energy ( ) is: Substitute and : Simplify: Step 4: Use the energy loss condition The ball loses of its kinetic energy, so the final kinetic energy is of the initial kinetic energy: Substitute and : Simplify: Multiply through by 8: Solve for : Take the square root: Step 5: Match with the options The coefficient of restitution is , which matches option (2). Final Answer: