Work and Energy Calculator
Choose what you want to compute — work, kinetic energy, or gravitational potential energy — and enter the values. The calculator shows the formula, the substitution, and the result.
Work is force times displacement in the direction of the force: \(W = Fd \cos\theta\). Kinetic energy is the energy of motion: \(KE = \tfrac{1}{2} m v^2\). Gravitational potential energy is the energy of position above a reference level: \(PE = mgh\). All three are measured in joules (J).
The Formulas
| Quantity | Formula | Units |
|---|---|---|
| Work | \(W = Fd \cos\theta\) | joules (J) |
| Kinetic energy | \(KE = \tfrac{1}{2} m v^2\) | joules (J) |
| Gravitational potential energy | \(PE = mgh\) | joules (J) |
The angle \(\theta\) in the work formula is the angle between the force vector and the displacement vector. If the force points in the same direction as the motion, \(\theta = 0\) and \(\cos\theta = 1\), so \(W = Fd\). If the force is perpendicular to the motion (like gravity on a horizontally-moving object), \(\theta = 90°\) and \(W = 0\) — no work done.
Worked Examples
Work
A horizontal force of \(20\) N pushes a box \(5\) m across the floor. The force and displacement are in the same direction.
\[W = Fd \cos\theta = (20)(5)\cos 0° = 100 \text{ J}\]
Kinetic energy
A \(2\) kg object moves at \(3\) m/s. Its kinetic energy is:
\[KE = \tfrac{1}{2}(2)(3)^2 = 9 \text{ J}\]
Potential energy
A \(5\) kg book sits on a shelf \(2\) m above the floor.
\[PE = mgh = (5)(9.8)(2) = 98 \text{ J}\]
This is the energy relative to the floor. If you'd taken the shelf itself as the reference (\(h = 0\) at the shelf), the PE would be \(0\) — potential energy always depends on where you put the zero.
Tips for Using the Calculator
- The work formula uses the angle between force and displacement, not the angle of either above horizontal. If the force is pushing horizontally and the object moves horizontally, \(\theta = 0\).
- Work-energy theorem: the net work done on an object equals the change in its kinetic energy: \(W_{\text{net}} = \Delta KE\). So if you know the work and the initial KE, you know the final KE.
- For kinetic energy, the sign of velocity doesn't matter — KE is always non-negative since it depends on \(v^2\).
- For potential energy, height is relative to whatever reference level you choose. Pick one and stay consistent.
Frequently Asked Questions
Why does work depend on the angle between force and displacement?
Only the component of the force along the direction of motion does work. If you push a sled forward (force horizontal) while gravity pulls it down (force vertical), gravity is perpendicular to the motion — \(\cos 90° = 0\) — so gravity does no work on the sled. The horizontal component of any applied force is what counts: \(F \cos\theta\) gets multiplied by the distance \(d\) to give the work.
What's the work-energy theorem?
The work-energy theorem states that the net work done on an object equals its change in kinetic energy:
\[W_{\text{net}} = \Delta KE = \tfrac{1}{2} m v^2 - \tfrac{1}{2} m v_0^2\]
It's a powerful shortcut: if you can compute the total work done by all forces, you immediately know how much the object sped up or slowed down without needing the kinematic equations.
Can kinetic energy be negative?
No. Kinetic energy depends on \(v^2\), which is always non-negative, so \(KE \geq 0\) for any object. A moving object always has positive KE; an object at rest has zero KE.
Can potential energy be negative?
Yes — and it's not unusual. PE is measured relative to a chosen reference height. If an object is below your reference level, its \(h\) is negative, so \(PE = mgh\) is negative. That just means it has less energy than at the reference level. Differences in PE are what matter physically; the absolute value depends on your choice of zero.
What about other forms of potential energy?
This calculator handles gravitational potential energy near Earth's surface, where the formula \(PE = mgh\) applies. Other potential energy formulas exist for different situations:
- Spring (elastic): \(PE = \tfrac{1}{2} k x^2\)
- Gravitational (far from Earth, general): \(PE = -G \dfrac{M m}{r}\)
- Electric (point charges): \(PE = k_e \dfrac{q_1 q_2}{r}\)
For introductory mechanics, the simple \(mgh\) form is what you need.