Question

A body is projected vertically upwards fron the sureface of earth wity a velocity equal to half escape speed

Answer 1

ed by it?

Since no answer yet has explicitly given the maximum height I will try my hand at this problem:

Let’s imagine an object of mass msubject to gravity from a planet with mass M. This object moves radially from a distance r0 to the center of the planet up to a distance r.

Since the force on the object is :

F=−GMmr2

The work done on the object by the gravitational field is:

W=∫rr0−GMmr2dr

W=GMm(1r−1r0) (1)

We also know that at a distance r0 from the center of the planet the escape velocity of the object is given by equation:

Ve=2GMr0−−−−√

Since the object is thrown with initial velocity equal to Ve2, then its kinetic energy is :

Ek=12mV2=GMm4r0 (2)

If this kinetic energy is enough to move the object to distance r , it must cancel the work done by the gravity field, so equations (1 ) and (2) add to zero:

GMm(1r−1r0)+GMm4r0=0

Simplifying this equation, we get:

1r=1r0−14r0

or

r=43r0

If we express this as

r=r0+13r0

We can say that the body reaches a height of one third its current distance to the center of the planet.