How Does Kick Dynamics Affect Soccer Ball Impulse and Force?

[xshezsciencex]

A player kicks a soccer ball (0.45kg). The player's foot is in contact with the ball for 3.0 H 10^-3 s and the force F(t)=[(6.0 * 10^6)t - (2.0 * 10^9)t^2 ]N for 0<_t<_3.0*10^-3s where t is in seconds.
Find the magnitude of (a) the impulse of the ball due to the kick, (b) the average and maximum force from the player's foot during the period of contact and (c) the ball's kinetic energy immediately after it loses contact with the player's foot.

 

[kuruman]

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[kerimek]

I would like to provide an analysis of the given content on magnitude and impulse. In this scenario, a player kicks a soccer ball with a mass of 0.45 kg. The player's foot is in contact with the ball for 3.0*10^-3 seconds and the force acting on the ball is given by F(t)=[(6.0 * 10^6)t - (2.0 * 10^9)t^2 ]N for 0<_t<_3.0*10^-3s where t is in seconds.

(a) The impulse of an object is defined as the change in its momentum. In this case, the momentum of the ball changes due to the kick from the player's foot. To find the magnitude of impulse, we need to find the change in momentum of the ball. The initial momentum of the ball is zero as it is at rest before the kick. The final momentum can be calculated using the formula p=mv, where p is the momentum, m is the mass of the ball and v is its velocity. As the ball is kicked, it acquires a velocity which can be calculated using the formula v=at, where a is the acceleration and t is the time. The acceleration can be calculated by taking the derivative of the given force function with respect to time, which gives a=6.0*10^6 - 4.0*10^9t. Plugging in the time of contact, t=3.0*10^-3s, we get a=6.0*10^6 - 12.0*10^3= -6.0*10^6 m/s^2. Now, substituting the values of mass and acceleration in the formula for momentum, we get the final momentum of the ball as p= 0.45*(-6.0*10^6)=-2.7*10^6 kgm/s. The change in momentum or the magnitude of impulse is therefore |-2.7*10^6| = 2.7*10^6 kgm/s.

(b) To find the average force from the player's foot during the period of contact, we can use the formula F=Δp/Δt, where Δp is the change in momentum and Δt is the change in time. As we have already calculated the change in momentum as 2.7