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. 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.

$\int _{2d}^{D}{\frac {GM_{1}M_{2}}{r^{2}}}dr=$ Gravitational Potential Energy

$\left[{\frac {-GM_{1}M_{2}}{r}}\right]_{2d}^{D}=-GM_{1}M_{2}\left[{\frac {1}{D}}-{\frac {1}{2d}}\right]={\frac {1}{2}}M_{2}v^{2}$

${\sqrt {-2GM_{1}\left[{\frac {1}{D}}-{\frac {1}{2d}}\right]}}=v=25,800ms^{-1}$

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

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

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.