A-level Physics (Advancing Physics)/Gravitational Potential Energy/Worked Solutions

1. A ball rolls down a 3m-high smooth ramp. What speed does it have at the bottom?

$mgh=\frac{1}{2}m{v}^2$

$gh=\frac{1}{2}{v}^2$

$v=\sqrt{2gh}=\sqrt{2\times 9.81\times 3}=7.67{\text{ ms}}^{-1}$

2. In an otherwise empty universe, two planets of mass 10²⁵ kg are 10¹² m apart. Both the planets have a radius of 10⁶ m. What are their speeds when they collide?

Let $M_1$ be the mass of planet 1 and $M_2$ be the mass of planet 2. Both are $1\times 10^{25}$ kg

Assume planet 1 to be stationary and planet 2 to be accelerating towards it (relative).

Let D = $1\times 10^{12}$ meters = distance between the center of the two planets. Let d = $1\times 10^6$ meters = radius of planets.

$\underset{2d}{\overset{D}{\int }}\frac{G{M}_1{M}_2}{r^2}dr=$ Gravitational Potential Energy

${\left[\frac{-G{M}_1{M}_2}{r}\right]}_{2d}^D=-G{M}_1{M}_2[\frac{1}{D}-\frac{1}{2d}]=\frac{1}{2}{M}_2{v}^2$

$\sqrt{-2G{M}_1[\frac{1}{D}-\frac{1}{2d}]}=v=25,800m{s}^{-1}$

3. What is the least work a 2000 kg car must do to drive up a 100m hill?

$mgh=2000\times 9.81\times 100=1.962\text{ MJ}$

4. How does the speed of a planet in an elliptical orbit change as it nears its star?

As it nears the star, it loses gravitational potential energy, and so gains kinetic energy, so its speed increases.