Gravitational potential energy and kinetic energy

Gravitational potential energy (GPE) and kinetic energy (KE) are two fundamental forms of mechanical energy. GPE is the energy an object has due to its position in a gravitational field, while KE is the energy due to its motion.

Gravitational Potential Energy (GPE)

When an object is raised in a uniform gravitational field (such as near the Earth's surface), work is done against gravity. This work is stored as gravitational potential energy.

Derivation of $\Delta E_p = mg\Delta h$

Work done to lift an object of mass $m$ by a vertical height $\Delta h$ against gravity:

• The force required to lift: $F = mg$ (weight of the object)
• The distance moved in the direction of the force: $s = \Delta h$

Work done, $W = F s = mg\Delta h$

This work increases the gravitational potential energy:

$$\Delta E_p = mg\Delta h$$

• $\Delta E_p$: change in gravitational potential energy (J)
• $m$: mass (kg)
• $g$: gravitational field strength (N kg$^{-1}$), typically $9.81$ N kg$^{-1}$
• $\Delta h$: change in height (m)

Kinetic Energy (KE)

Kinetic energy is the energy an object has due to its motion.

Derivation of $E_k = \frac{1}{2}mv^2$

Consider an object of mass $m$ accelerating from rest under a constant force $F$ over a distance $s$:

• Work done by the force: $W = F s$
• From Newton's second law: $F = ma$
• From equations of motion (starting from rest): $v^2 = u^2 + 2as$, with $u = 0$, so $v^2 = 2as \implies s = \frac{v^2}{2a}$

Substitute $F$ and $s$:

$$W = F s = ma \cdot \frac{v^2}{2a} = \frac{1}{2}mv^2$$

This work done increases the kinetic energy:

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

• $E_k$: kinetic energy (J)
• $m$: mass (kg)
• $v$: speed (m s$^{-1}$)

Summary

• Gravitational potential energy change: $\Delta E_p = mg\Delta h$
• Kinetic energy: $E_k = \frac{1}{2}mv^2$

Both GPE and KE are scalar quantities and are measured in joules (J).