# 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?

$mgh = \frac{1}{2}mv^2$

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

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

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

$\frac{GMm}{0.5r} = \frac{1}{2}mv^2$

$\frac{GM}{0.5r} = \frac{1}{2}v^2$

$\frac{4GM}{r} = v^2$

$v = 2\sqrt{\frac{GM}{r}} = 2\sqrt{\frac{6.67 \times 10^{-11} \times 10^{25}}{10^{12}}} = 52\mbox{ ms}^{-1}$

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

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

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.