A tennis racket hits a tennis ball with a force of F=at−bt2, where a = 1200 N/ms , b = 370 N/ms2 , and t is the time (in milliseconds). The ball is in contact with the racket for 2.70 ms . If the tennis ball has a mass of 64.9 g , what is the resulting velocity of the ball, v, after the ball is hit by the racket? Express your answer numerically in meters per second to three significant fi

Given Information:  

Mass of tennis ball = m = 64.9 g

Contact time = Δt = 2.70 ms

Force = F = at - bt²

a = 1200 N/ms

b = 370 N/ms²

Assuming that a and b is given as newton per milliseconds

Required Information:  

Velocity of the ball = Vf = ?  

Vf = 22.496 m/s

Explanation:  

The change in momentum of the ball is given by

Δp = -FΔt

Lets first find out the force acting on the tennis ball

F = aΔt - bΔt²

F = (1200)(2.70) - (370)(2.70)²

F = 542.7 N

Δp =  542.7*0.0027

Δp = 1.46 N.s

Δp = mVf - mVi

initial velocity is zero

Vf = Δp/m

Vf = 1.46/0.0649

Vf = 22.496 m/s

Therefore, the speed of the ball is 22.496 m/s