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.

$$\int_{2d}^{D}\frac{GM_1M_2}{r^2}dr = $$ Gravitational Potential Energy

$$ \left [ \frac{-GM_1M_2}{r}\right ]_{2d}^{D} = -GM_1M_2 \left [\frac{1}{D}-\frac{1}{2d} \right] = \frac{1}{2}M_2v^2 $$

$$\sqrt{-2GM_1 \left [\frac{1}{D}-\frac{1}{2d} \right]} = v = 25,800ms^{-1}$$

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.