Energy is the ability to do work. The SI unit for energy is the Joule ($J$).
Types of energy
There are many types of energy:
- Light (electromagnetic) energy
- Sound energy
- Thermal (heat) energy
- Mechanical energy (potential and kinetic energy)
Kinetic energy
This is the energy a body possesses by virtue of its motion. The kinetic energy of a body with mass $m$ and velocity $v$ is given as: $$ \begin{equation}\begin{aligned} KE=\frac12 mv^2\\ \end{aligned}\end{equation} $$
A change in kinetic energy for a body with initial velocity $u$ and final velocity $v$ is: $$ \begin{equation}\begin{aligned} \Delta KE&=KE_f-KE_i\\ &=\frac12 mv^2-\frac12 mu^2\\ \end{aligned}\end{equation} $$
Potential energy
This is the energy a body possesses by virtue of its state (e.g. chemical potential energy) or position within a field (e.g. gravitational potential energy and electric potential energy).
The gravitational potential energy of a body with mass $m$ at a height of $h$ above the reference level (e.g. the ground) is: $$ \begin{equation}\begin{aligned} GPE=mgh\\ \end{aligned}\end{equation} $$
The change in GPE if the body falls from an initial height of $h_i$ to a final height of $h_f$ is: $$ \begin{equation}\begin{aligned} \Delta GPE&=GPE_f-GPE_i\\ &=mgh_f-mgh_i\\ &=mg(h_f-h_i)\\ &=mg\Delta h\\ \end{aligned}\end{equation} $$
Example: Find the change in gravitational potential energy of an object whose mass is $3kg$ and falls from a height of $7m$ to a height of $1m$.
Solution: $$ \begin{equation}\begin{aligned} \Delta GPE&=mg\Delta h\\ &=3kg\times 9.81ms^{-2}\times (1m - 7m)\\ &=-176.58J\\ \end{aligned}\end{equation} $$
Law of conservation of energy
Energy cannot be created nor destroyed, it can only be transformed from one form to the other.
Consider the conversion of gravitational potential energy to kinetic energy:
$$ \begin{equation}\begin{aligned} GPE_{top}&=KE_{bottom}\\ mgh&=\frac12 mv^{2}\\ gh&=\frac12 v^{2}\\ v^2&=2gh\\ v&=\sqrt{2gh}\\ \end{aligned}\end{equation} $$Find the final velocity of an object, mass $5kg$, falling from a height of $8m$ to the ground. $$ \begin{equation}\begin{aligned} v&=\sqrt{2gh}\\ &=\sqrt{2\times 9.81ms^{-2} \times 8m}\\ &=12.53ms^{-1}\\ \end{aligned}\end{equation} $$
Work
This is the dot product of the force applied to an object and the displacement of the object: $$ \begin{equation}\begin{aligned} Work=F\cdot s\\ \end{aligned}\end{equation} $$
If the displacement is in the direction of the force then work is done on the object and if the displacement is opposite in the direction of the force, then work is done by the object.
Power and velocity
Power is the rate of change of energy: $$ \begin{equation}\begin{aligned} P=\frac{E}{t}\\ \end{aligned}\end{equation} $$
We know that energy is the ability to do work: $$ \begin{equation}\begin{aligned} P=\frac{\text{Work}}{t}\\ \end{aligned}\end{equation} $$
Work is the product of force and displacement: $$ \begin{equation}\begin{aligned} P&=\frac{F\cdot s}{t}\\ &=F\cdot \frac{s}{t}\\ &=F\cdot v\\ \end{aligned}\end{equation} $$