Question

a ball with mass 0.15 kg is thrown upward with initial velocity 20 m/sec from the roof of a building 30 m high. there is a force due to air resistance of |v|/30, where velocity v is measured in m/sec.
a. find the maximum height above the ground the ball reaches.
b. find the time the ball hits the ground.

you cannot use the kinematic equations.

Answer 1

This IS a kinematics problem. Any equation used to solve it involves kinematics. An exact solution would require solving a (kinematic) differential equation.

A good approximation to part (a) can be obtained by assuming an average air resistance force of |V|/60 , where V is the initial velocity of 20 m/s. Thus work done against friction going up is
V/60*H, and initial kientic energy equals potential energy gain at the highest elevation PLUS work done against friction.
(1/2) M V^2 = M g H + V/60*H
H = (1/2)V^2/[g + V/(60M)]= (1/2)V^2/11
= 18.2 m

The time spent by the ball going up is
very nearly H/(V/2) = 0.46 s

Use a similar energy conservation argument to estimate the (longer) time it takes for the ball to come back down. Note that it must travel an additional 30 m to reach the ground.

Answer 2

how would you solve it with differential equations

Answer 3

Solve
M dv/dt = -M g -v/60*H
(v>0)

followed by
M dv/dt = -M g + v/60*H
(v<0)

Once you have v(t), integrate v dt to get the distance travelled vs t.