The equation is .

Where is the objects distance above the surface at time .

is the radius of earth, is the mass of the object  and is acceleration.

From newtons second law: .

Therefore, .

(a) Show that .

The velocity of the object is .

.

Using chain rule : .

Since .

Multiply each side by .

Susbtitute .

Integrate both sides.

The initial velocity is and maximum height is .

Rewrite the above integral .

\ \

.

Therefore, the initial velocity is .

(b)

The initial velocity is .

Find the escape velocity .

Substitute .

Therefore, the escape velocity .

(c)

The escape velocity .

Substitute and .

Since .

Hence, .

.

Therefore, the escape velocity is and .

(a) The initial velocity is .

(b) The escape velocity .

(c) The escape velocity is and .