Rafael drops a hard rubber hall, of mass m = 295 g. onto the pavement Ile drops

Question

Rafael drops a hard rubber hall, of mass m = 295 g. onto the pavement Ile drops it from a height h_1 = 2.7 m. and it bounces hack to a height h_2=2.3 m. While the hall is in contact with the ground, the ground exerts a non constant force on the ball as shown in the figure. During the time interval Delta t_1, the force rises linearly to F_max, and during the time interval Delta t_2 the force returns linearly to zero as the ball leaves the ground. Refer to the figure. Randomized Variables m = 295 g h_1 = 2.7 m h_2 = 2.3 m Enter an expression for the magnitude of the impulse delivered to the ball by the ground, in terms of F_max and the time interval Delta t_1 and Delta t_2. Select from the variables below to write your expression. alpha, beta, Delta t_1, Delta t_2, theta, a, d, F_max, g, h, j, k, m, p. t For the time intervals Delta t_1 = 2.5 ms and Delta t_2 = 4.5, what is the magnitude of the maximum force between the ground and the ball in newtons?

Explanation / Answer

part a:

As we all know that the magnitude of the impulse is given by the area under curve F vs t.

Impulse = 0.5*(delta_t1+delta_2)*F_max

part b:

Consider the collision:

velocity before the collision v1 = sqrt(2gh) = sqrt(2*9.81*2.7) = 7.278 m/s

velocity after the collision v2 = sqrt(2gh) = sqrt(2*9.81*2.3) = 6.718 m/s

change in momentum = m*delta_v = (295/1000)*(7.278-6.718) = 0.1652 kg.m/s

The impulse is change in momentum.

So, 0.5*(2.5+4.5)*F_max/1000 = 0.1652

F_max = 165.2*2/7 = 47.2 N

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