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

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

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

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

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

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

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

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

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

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

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

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

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.