We can solve this problem by using the work-energy principle, which states that the work done on an object is equal to its change in kinetic energy.
The initial kinetic energy of the soccer ball is:
K_i = (1/2) * m * v_i^2
where m is the mass of the ball, and v_i is its initial velocity.
The final kinetic energy of the soccer ball is zero, since it comes to a stop.
Therefore, the work done on the ball by Sarah is:
W = K_f - K_i = -K_i
where K_f is the final kinetic energy (which is zero), and the negative sign indicates that the work done by Sarah is negative (i.e., she does negative work to stop the ball).
The average net force exerted by Sarah on the ball is given by:
F = W / d
where d is the distance over which Sarah applies the force (i.e., the distance over which she stops the ball).
a) If it takes Sarah 0.4 seconds to stop the ball, the distance over which she applies the force is:
d = v_i * t = 8 m/s * 0.4 s = 3.2 m
The initial kinetic energy of the ball is:
K_i = (1/2) * m * v_i^2 = (1/2) * 0.4 kg * (8 m/s)^2 = 12.8 J
Therefore, the work done by Sarah is:
W = -K_i = -12.8 J
The average net force exerted by Sarah on the ball is:
F = W / d = (-12.8 J) / (3.2 m) = -4 N
Therefore, the average net force exerted by Sarah on the ball if it took her 0.4 seconds to stop it is -4 N.
b) If it takes Sarah 0.6 seconds to stop the ball, the distance over which she applies the force is:
d = v_i * t = 8 m/s * 0.6 s = 4.8 m
The initial kinetic energy of the ball is still:
K_i = (1/2) * m * v_i^2 = (1/2) * 0.4 kg * (8 m/s)^2 = 12.8 J
Therefore, the work done by Sarah is:
W = -K_i = -12.8 J
The average net force exerted by Sarah on the ball is:
F = W / d = (-12.8 J) / (4.8 m) = -2.67 N
Therefore, the average net force exerted by Sarah on the ball if it took her 0.6 seconds to stop it is -2.67 N.
