A projectile of mass m is thrown vertically up with an initial velocity v from the surface of earth (mass of earth = M). If it comes to rest at a height h, the change in its potential energy is

Options

A. `GMmh//R(R+h)`
B. `GMmh^(2)//R(R+h)^(2)`
C. `GMmhR//R(R+h)`
D. `GMm//hR(R+h)`

Answer

To find the change in potential energy of a projectile of mass \( m \) thrown vertically upwards from the surface of the Earth, we can follow these steps:

### Step 1: Understand the potential energy formula
The gravitational potential energy (PE) between two masses \( m \) and \( M \) (where \( M \) is the mass of the Earth) at a distance \( r \) from the center of the Earth is given by the formula:
\[ PE = -\frac{G M m}{r} \]
where \( G \) is the gravitational constant, and \( r \) is the distance from the center of the Earth.

### Step 2: Determine the initial potential energy (PE1)
At the surface of the Earth, the distance \( r \) is equal to the radius of the Earth, denoted as \( R \). Therefore, the initial potential energy \( PE_1 \) when the projectile is at the surface is:
\[ PE_1 = -\frac{G M m}{R} \]

### Step 3: Determine the final potential energy (PE2)
When the projectile reaches a height \( h \), the distance from the center of the Earth becomes \( R + h \). The potential energy \( PE_2 \) at this height is:
\[ PE_2 = -\frac{G M m}{R + h} \]

### Step 4: Calculate the change in potential energy
The change in potential energy \( \Delta PE \) is given by:
\[ \Delta PE = PE_2 - PE_1 \]
Substituting the expressions for \( PE_1 \) and \( PE_2 \):
\[ \Delta PE = \left(-\frac{G M m}{R + h}\right) - \left(-\frac{G M m}{R}\right) \]
This simplifies to:
\[ \Delta PE = -\frac{G M m}{R + h} + \frac{G M m}{R} \]
Factoring out \( G M m \):
\[ \Delta PE = G M m \left(\frac{1}{R} - \frac{1}{R + h}\right) \]

### Step 5: Simplify the expression
To combine the fractions:
\[ \Delta PE = G M m \left(\frac{(R + h) - R}{R(R + h)}\right) \]
This further simplifies to:
\[ \Delta PE = G M m \left(\frac{h}{R(R + h)}\right) \]

### Final Result
Thus, the change in potential energy when the projectile comes to rest at height \( h \) is:
\[ \Delta PE = \frac{G M m h}{R(R + h)} \]