Gravitational Field, Potential Energy and Escape Speed: Quick Revision

A brief note on Gravitational Field, Potential Energy and Escape Speed. Read it before a test. Try each question at the end before you open its answer.

The big ideas

• Field: the pull on each kilogram, $\dfrac{GM}{r^2}$, pointing to the centre.
• Potential energy: $U = -\dfrac{GMm}{r}$, with zero far away. It is negative because gravity holds the body in.
• Potential: $V = -\dfrac{GM}{r}$, the potential energy of each kilogram. Potentials add as plain numbers; fields add as vectors.
• Shells: a uniform shell pulls an outside body as if its mass sat at its centre. Inside it, the field is zero and the potential is the same everywhere.
• Escape speed: $v_e = \sqrt{\dfrac{2GM}{R}} = \sqrt{2gR}$, about $11.2\ \text{km/s}$ for the Earth. It does not depend on the body's mass or on the direction of launch.

The facts at a glance

• Field outside: $\dfrac{GM}{r^2}$
• Potential outside: $-\dfrac{GM}{r}$
• Work from $r_1$ to $r_2$: $GMm\left(\dfrac{1}{r_1} - \dfrac{1}{r_2}\right)$
• Near the Earth: $GM = gR^2$
• Escape speed: $\sqrt{2gR}$
• Fired up at $v$: $h = \dfrac{Rv^2}{v_e^2 - v^2}$

A worked example

Ganymede, Jupiter's largest moon, has $g = 1.43\ \text{m/s}^2$ at its surface and a radius of $2.63 \times 10^{6}\ \text{m}$. A $60\ \text{kg}$ instrument rests there.

1. Potential at the surface. $V = -gR = -3.76 \times 10^{6}\ \text{J/kg}$.
2. Work to take the instrument far away. $m \times gR = 60\ \text{kg} \times 3.76 \times 10^{6}\ \text{J/kg} = 2.26 \times 10^{8}\ \text{J}$.
3. Escape speed. $\sqrt{2gR} = \sqrt{2 \times 3.76 \times 10^{6}\ \text{J/kg}} = 2.74\ \text{km/s}$.

Practice with new numbers

A program to try

The program works out the worked example.

Mistakes to avoid

• A positive potential energy. With zero far away, it is always negative.
• $mgh$ for big heights. Use $GMm\left(\dfrac{1}{r_1} - \dfrac{1}{r_2}\right)$.
• Zero field means zero potential. Inside a shell the field is zero, but the potential is $-\dfrac{GM}{a}$.
• Subtracting speeds. Launched faster than $v_e$, a body keeps $\sqrt{v^2 - v_e^2}$, not $v - v_e$.
• The height for the distance from the centre. At a height $h$, $r = R + h$.

Check yourself

1. What is the potential energy of balls of $3.0\ \text{kg}$ and $5.0\ \text{kg}$ that are $0.50\ \text{m}$ apart?

2. What is the gravitational field inside a hollow uniform shell?

3. A planet has twice the Earth's mass and twice its radius. What is its escape speed?

4. Does the escape speed depend on the angle of launch?

5. Why do rockets launch eastward, and from near the equator if they can?