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

An object of mass $m$ is dropped from height $h$ above a planet of mass $M$ and radius $R .$ Find an expression for the object’s speed as it hits the ground.

Short Answer

Expert verified

the expression for object's speed is $v = \sqrt{2 G M (\frac{1}{R + h} - \frac{1}{R})}$

Step by step solution

Given information

mass of an object is $m$

height of an object to be dropped is $h$

mass of the planet $M$

radius of the planet $R$

Explanation

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

$E = U = \frac{G M m}{(R + h)}$

where

$G$ is the gravitational constant

$M$ is the mass of the planet

$m$ is the mass of the object

$R$ is the radius of the planet

$h$ is the altitude of the object

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

$E = \frac{G M m}{R} + \frac{1}{2} m v^{2}$

where

$v$ is the speed of the object at the ground

Since the mechanical energy is conserved, we can write

$\frac{G M m}{R} + \frac{1}{2} m v^{2} = \frac{G M m}{R + h}$

and solving for $v$ , we find

$v^{2} = 2 G M (\frac{1}{R + h} - \frac{1}{R}) v = \sqrt{2 G M (\frac{1}{R + h} - \frac{1}{R})}$

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An object of mass $m$ is dropped from height $h$ above a planet of mass $M$ and radius $R .$ Find an expression for the object’s speed as it hits the ground.

Short Answer

Step by step solution

Given information

Explanation

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

Short Answer

Step by step solution

Given information

Explanation

One App. One Place for Learning.

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An object of mass $m$ is dropped from height $h$ above a planet of mass $M$ and radius $R .$ Find an expression for the object’s speed as it hits the ground.