Physics Center of Mass and Linear Momentum?

Question

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 3.0 x 10^-3 seconds, and the force of the kick is given by:
F(t) = [(6.0 x10^6)t - (2.0 x10^9)t^2] N
for 0 <_ t <_ 3.0 x10^-3 seconds, where t is in seconds. Find the magnitudes of:
(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.
(D) The ball's velocity immediately after it loses contact with the player's foot.

The answers are:
(A) 9.0 kg x m/s
(B) 3.0 kN
(C) 4.5 kN
(D) 20 m/s

But I don't know how to get these answers, please help Me!

Answer 1

(A) The impulse on the ball due to the kick is the integral of the force of the time. Let a = 6x10^6 and b = 2x10^9, then

Impulse = integral(F(t))dt = integral(a*t - b*t^2)dt
= (0.5*a*t^2 - b*t^3/3) | (3x10-3,0)

where the last part indicate evaluated from 0 to 3x10^-3 seconds.

(B) The average force is the same integral divided by the time interval, or the impulse divided by the time over which the impulse is applied.

Impulse/dT = Impulse/(3x10^-3 - 0)

(C) The time of maximum force can be found by taking the derivative of the force and looking at which time the derivative equals zero.

dF/dt = a - 2*b*tmax = 0

This leads to

tmax = a/(2*b)

Using this time in the force equation will give you the maximum force on the ball.

(D) The average force can be expressed as

= m*dV/dT

where dV is the change in velocity and dT is the time over which the force is applied. Solving for dV

dV = *dT/m

will give the speed immediately after the impulse force.

Answer 2

The system is not influenced by the outside world so for an outside observer, the system will not change, in other words, the position of its center of mass will not be altered. Given the fact that the two skaters will eventually meet and in light of what I've already said, they have no choice but too meet in the system's center of mass. let x be the initial distance between the 40kg skater and the center of mass and l=10m, the length of the pole, then l-x is distance between the 65kg skater in the center of mass: x *40kg = (l-x) * 65kg x* 105= l*65 x= 650/105 m = 6.19m The 40kg skater travels the length of x, the initial distance between him and their center of gravity, he travels aproximately 6.19m