recall and use the formula ∆EP = mg∆h for gravitational potential energy changes in a uniform gravitational field

1. Introduction

In this topic we examine how the energy of a body changes when it moves in a uniform gravitational field. The key formula to remember is

\$\Delta E_{\text{P}} = m g \,\Delta h\$

where \$m\$ is the mass of the object, \$g\$ is the acceleration due to gravity (≈ 9.81 m s⁻² near the Earth’s surface) and \$\Delta h\$ is the vertical displacement.

2. Gravitational Potential Energy (GPE)

2.1 Definition

Gravitational potential energy is the energy stored in an object because of its position in a gravitational field. In a uniform field the change in GPE when the object moves vertically is given by the formula above.

2.2 Symbol and Units

2.3 Using the Formula

• Identify the mass \$m\$ of the object.
• Determine the vertical displacement \$\Delta h\$ (positive when the object rises, negative when it falls).
• Apply \$g = 9.81\ \text{m s}^{-2}\$ unless otherwise specified.
• Calculate \$\Delta E_{\text{P}} = m g \Delta h\$.

3. Kinetic Energy (KE)

3.1 Definition

The kinetic energy of a body moving with speed \$v\$ is

\$E_{\text{K}} = \frac{1}{2} m v^{2}\$

3.2 Relationship with GPE

When only gravity does work, the mechanical energy is conserved:

\$E{\text{P,initial}} + E{\text{K,initial}} = E{\text{P,final}} + E{\text{K,final}}\$

This principle allows us to link changes in height to changes in speed.

4. Worked Example

Problem: A 2.0 kg ball is dropped from a height of 5.0 m. Find its speed just before it hits the ground, neglecting air resistance.

1. Initial kinetic energy \$E_{\text{K,i}} = 0\$ (the ball starts from rest).
2. Initial potential energy \$E_{\text{P,i}} = m g h = (2.0)(9.81)(5.0) = 98.1\ \text{J}\$.
3. At the ground \$h = 0\$, so \$E_{\text{P,f}} = 0\$.
4. Conservation of mechanical energy gives \$E{\text{K,f}} = E{\text{P,i}} = 98.1\ \text{J}\$.
5. Set \$\frac{1}{2} m v^{2} = 98.1\$ and solve for \$v\$:

   \$v = \sqrt{\frac{2E_{\text{K,f}}}{m}} = \sqrt{\frac{2(98.1)}{2.0}} \approx 9.9\ \text{m s}^{-1}\$

The ball’s speed just before impact is approximately \$9.9\ \text{m s}^{-1}\$.

5. Common Mistakes to Avoid

• Using \$\Delta h\$ with the wrong sign – remember that upward displacement is positive.
• Confusing \$g\$ with \$9.8\ \text{m s}^{-2}\$ versus \$9.81\ \text{m s}^{-2}\$; the exam will specify which value to use.
• Omitting the factor of \$m\$ when calculating \$\Delta E_{\text{P}}\$.
• Assuming mechanical energy is conserved when non‑conservative forces (e.g., friction) are present.

6. Summary Checklist

1. Write down the known values: \$m\$, \$g\$, \$\Delta h\$.
2. Apply \$\Delta E_{\text{P}} = m g \Delta h\$.
3. If kinetic energy is involved, use \$E_{\text{K}} = \frac{1}{2} m v^{2}\$.
4. Use energy conservation to relate GPE and KE when appropriate.
5. Check units and sign of \$\Delta h\$ before finalising the answer.

7. Practice Questions

1. A 5.0 kg crate is lifted vertically by 3.2 m. Calculate the increase in its gravitational potential energy.
2. A 0.50 kg stone is thrown upward with an initial speed of 12 m s⁻¹. What is the maximum height it reaches above the launch point? (Take \$g = 9.81\ \text{m s}^{-2}\$.)
3. A 1.2 kg ball rolls down a 4.0 m high frictionless ramp. Determine its speed at the bottom.