Final answer:
To find the coefficient of restitution, use the formula c=(h/H)^(1/2), where h is the height to which the ball bounces and H is the height from which it is dropped. After the third bounce, the coefficient is approximately 0.7. To find the speed with which the ball comes up from the ground after the third bounce, use the principle of conservation of mechanical energy. The speed is approximately 6.9 m/s.
Step-by-step explanation:
To calculate the coefficient of restitution, we can use the formula c = (h/H)^(1/2), where h is the height to which the ball bounces and H is the height from which the ball is dropped. Let's look at the given information. The ball is dropped from a height of 5m and bounces repeatedly, with 64% of the energy being retained during each bounce. So, after the first bounce, the ball reaches a height of 0.64*5 = 3.2m. After the second bounce, it reaches a height of 0.64*3.2 = 2.048m. And after the third bounce, it reaches a height of 0.64*2.048 = 1.31072m. Therefore, the coefficient of restitution can be calculated as c = (1.31072/5)^(1/2) ≈ 0.7.
To find the speed with which the ball comes up from the ground after the third bounce, we can use the principle of conservation of mechanical energy. The mechanical energy at the topmost point of the ball's trajectory will be equal to the mechanical energy at the initial height. Let's consider the initial height of 5m. At this height, the ball has gravitational potential energy given by mgh, where m is the mass of the ball (100g = 0.1kg), g is the acceleration due to gravity (9.8m/s^2), and h is the height (5m). So, the initial gravitational potential energy is 0.1*9.8*5 = 4.9J. This energy will be converted to kinetic energy at the topmost point. Therefore, 4.9J = 0.5mv^2, where v is the velocity at that point. Solving for v gives us v = sqrt((2*4.9)/0.1) ≈ 9.9m/s. Since the coefficient of restitution is given as 0.7, the velocity with which the ball comes up from the ground after the third bounce will be 0.7*9.9 ≈ 6.9m/s.