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Chapter 13: Q.42 - Excercises And Problems (page 355)

An object of mass mis dropped from height habove a planet of mass Mand radius R.Find an expression for the object鈥檚 speed as it hits the ground.

Short Answer

Expert verified

the expression for object's speed isv=2GM1R+h-1R

Step by step solution

01

Given information

mass of an object ism

height of an object to be dropped is h

mass of the planet M

radius of the planetR

02

Explanation

The initial mechanical energy of the object, when it is located at height h above the the planet, is just gravitational potential energy:

E=U=GMm(R+h)

where

Gis the gravitational constant

Mis the mass of the planet

mis the mass of the object

Ris the radius of the planet

his the altitude of the object

When the object hits the ground, its mechanical energy will sum of potential energy and kinetic energy:

E=GMmR+12mv2

where

vis the speed of the object at the ground

Since the mechanical energy is conserved, we can write

GMmR+12mv2=GMmR+h

and solving for v, we find

v2=2GM1R+h-1Rv=2GM1R+h-1R

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