PhysicsIntroductory PhysicsHigh school

Work and Energy

Work, kinetic and potential energy, conservation of energy.

Work is done when a force moves an object: $W = Fd\cos\theta$ , where $\theta$ is the angle between force and displacement. Energy is the capacity to do work. Kinetic energy $\frac{1}{2}mv^2$ belongs to moving objects; gravitational potential energy $mgh$ depends on height.

The work-energy theorem says the net work on an object equals its change in kinetic energy. When only conservative forces such as gravity act, mechanical energy is conserved, so $KE + PE$ stays constant. This often solves motion problems far faster than forces and kinematics, because it ignores the path.

Key terms

Kinetic energy: Energy of motion.
$K = \tfrac{1}{2}mv^2$
Work: Force along the displacement times distance.
$W = Fd\cos\theta$

Key ideas

Units: joules, $1\text{ J} = 1\text{ N} \cdot \text{m}$ .
Forces perpendicular to motion, like the normal force on a level floor, do no work.
Conservation: $\frac{1}{2}mv_1^2 + mgh_1 = \frac{1}{2}mv_2^2 + mgh_2$ when friction is negligible.
Power is the rate of work: $P = \frac{W}{t}$ , in watts.

Worked example

A 2 kg ball is dropped from 5 m. How fast is it moving just before it lands (ignore air resistance)?

Start: $PE = mgh = 2 \times 9.8 \times 5 = 98$ J; $KE = 0$ .
End: all 98 J is kinetic, so $\frac{1}{2}(2)v^2 = 98$ .
$v^2 = 98$ , $v \approx 9.9$ m/s.
The mass cancels, so any mass lands at the same speed.

Answer: about $9.9$ m/s

Common mistakes

Forgetting to square the speed in $\frac{1}{2}mv^2$ .
Counting work by a force perpendicular to the motion.
Assuming energy is conserved when friction acts.

Practice questions

Try each one, then open the answer. Numerical answers on this page are verified by an automated calculation check.

1.Find the kinetic energy of a 3 kg object at 4 m/s.
Show answer
$24$ J
2.How much work lifts a 10 kg box 2 m (use $g = 9.8$ )?
Show answer
$196$ J
3.A 600 J task takes 12 s. What is the power?
Show answer
$50$ W

Key terms

Key ideas

Worked example

Common mistakes

Practice questions

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