A soccer player kicks a soccer ball of mass 045 kg that is i

A soccer player kicks a soccer ball of mass 0.45 kg that is initially at rest. The player\'s foot is in contact with the ball for 1.1 x 10-3 s, and the force of the kick is given by F(t) = [(7.7 x 104)t - (1.9 x 106)t2] N for 0 t 1.1 x 10-3s, where t is in seconds. Find the magnitudes of the following: (a) the impulse on the ball due to the kick, (b) the average force on the ball from the player\'s foot during the period of contact, (c) the maximum force on the ball from the player\'s foot during the period of contact, and (d) the ball\'s speed immediately after it loses contact with the player\'s foot.

Solution

Well, clearly they want you to integrate the force over the time integral. Yuck.
F(t)t = [(7.7 x 10^4)*t - (1.9 x 10^6)*t²] t·N = [(3.85X 10^4)t^2 - (0.633 X 10^6)t^3] N·s
evaluated at the limits t=0 &t=1.1*10^-3 is
(17.91 - 10.65) N·s - 0 =0.04574 N·s (a)

(b) average force is then impulse / time, or
F = 0.04574N·s / 0.0011s = 45.74N

(c) maximum force occurs where F/t = 0, or
7.1*10^4 = 3.8**10^6t
t = 0.019s
substitute into F(t) to find Fmax. (b)

(d) impulse = m·v, so
v = 0.04574N·s / 0.45kg = 0.1016m/s