# Work and energy

Canonical: https://duckyhelper.com/learn/physics/work-and-energy/
Updated: 2026-10-01

Work is energy transferred by a force acting over a distance: \(W = Fd\cos\theta\), measured in joules (J). Kinetic energy is energy of motion, \(KE = \tfrac{1}{2}mv^2\). Gravitational potential energy is energy stored by height, \(PE = mgh\). The work-energy theorem says the net work on an object equals its change in kinetic energy. With no friction, kinetic plus potential energy stays constant.

## Key ideas

$$
W = F d \cos\theta
$$

- \(W\) is work, in joules (J). \(1\ \text{J} = 1\ \text{N·m}\).
- \(F\) is the size of the force, in newtons (N), and \(d\) is the distance moved, in meters (m).
- \(\theta\) is the angle between the force and the direction of motion. Force along the motion gives positive work, against it gives negative work, and at 90° the work is zero.

$$
KE = \tfrac{1}{2} m v^2 \qquad PE_g = m g h
$$

- \(KE\) is kinetic energy and \(PE_g\) is gravitational potential energy, both in joules (J).
- \(m\) is mass (kg), \(v\) is speed (m/s), \(g = 9.8\ \text{m/s}^2\) and \(h\) is height above a level you choose as zero (m).

$$
W_{\text{net}} = \Delta KE \qquad KE_1 + PE_1 = KE_2 + PE_2 \ \text{(no friction)}
$$

**Power** is how fast work is done: \(P = W/t\), in watts (W), where \(1\ \text{W} = 1\ \text{J/s}\).

## Worked examples

**Example 1: work with a force at an angle**

Problem: You pull a sled 4.00 m across flat snow with a rope. The rope pulls with 50.0 N at 30.0° above the ground. How much work does the rope do?

1. Only the part of the force along the motion does work.

   $$
   W = Fd\cos\theta = (50.0\ \text{N})(4.00\ \text{m})\cos 30.0^\circ
   $$
2. Calculate.

   $$
   W = 200.\ \text{J} \times 0.866 = 173.2\ \text{J}
   $$

Answer: 173 J

**Example 2: kinetic energy**

Problem: What is the kinetic energy of a 0.50 kg ball moving at 12 m/s?

1. Square the speed first.

   $$
   KE = \tfrac{1}{2}(0.50\ \text{kg})(12\ \text{m/s})^2 = \tfrac{1}{2}(0.50)(144)\ \text{J}
   $$
2. Multiply.

   $$
   KE = 36\ \text{J}
   $$

Answer: 36 J

**Example 3: conservation of energy on a roller coaster**

Problem: A coaster car starts from rest at the top of a 45.0 m drop. Ignoring friction, how fast is it going at the bottom?

1. All the potential energy at the top becomes kinetic energy at the bottom.

   $$
   mgh = \tfrac{1}{2}mv^2
   $$
2. Mass cancels, so the answer does not depend on how heavy the car is.

   $$
   v = \sqrt{2gh} = \sqrt{2(9.8\ \text{m/s}^2)(45.0\ \text{m})} = 29.70\ \text{m/s}
   $$

Answer: 29.7 m/s

**Example 4: power climbing stairs**

Problem: A 60.0 kg student runs up stairs 4.50 m high in 6.00 s. What power does the student put out?

1. Work against gravity equals the gain in potential energy.

   $$
   W = mgh = (60.0)(9.8)(4.50) = 2646\ \text{J}
   $$
2. Divide by time.

   $$
   P = \frac{2646\ \text{J}}{6.00\ \text{s}} = 441\ \text{W}
   $$

Answer: 441 W

## Common mistakes and how to fix them

- **Counting work when nothing moves.** Holding a heavy box still does no work on it, because \(d = 0\).
- **Forgetting to square the speed.** Doubling the speed makes KE four times bigger, not twice. Fix: square \(v\) first.
- **Using conservation of energy with friction.** Friction turns some energy into heat. Fix: include the friction work, or only use \(KE + PE\) = constant when friction is ignored.
- **Forgetting the angle.** Fix: \(\cos\theta\) uses the angle between force and motion. A force at right angles to the motion does zero work.

**Practice problems**

1. How much work does it take to lift a 2.0 kg book 1.5 m at a steady speed?
   A. 3.0 J
   B. 29 J
   C. 15 J
   D. 2.9 J

   Answer: 29 J. The lifting force equals the weight, \(2.0 \times 9.8 = 19.6\) N. \(W = 19.6 \times 1.5 = 29.4\) J, which is 29 J.

2. If a car's speed doubles, what happens to its kinetic energy?
   A. It doubles
   B. It stays the same
   C. It becomes 4 times as large
   D. It halves

   Answer: It becomes 4 times as large. KE depends on \(v^2\), and \(2^2 = 4\).

3. A ball is dropped from 5.0 m. Ignoring air resistance, how fast is it moving just before it hits the ground?

   Answer: 9.9 m/s. \(v = \sqrt{2gh} = \sqrt{2 \times 9.8 \times 5.0} = \sqrt{98} = 9.90\), which is 9.9 m/s.

4. A box slides across a flat floor. How much work does the normal force do on it?
   A. 0 J
   B. \(mgd\)
   C. \(-mgd\)
   D. It depends on the friction

   Answer: 0 J. The normal force points up and the motion is sideways, so \(\theta = 90^\circ\) and \(\cos 90^\circ = 0\).

5. A 1200 kg car speeds up from 10. m/s to 20. m/s. How much net work was done on it?

   Answer: \(1.8 \times 10^5\) J. \(W_{\text{net}} = \Delta KE = \tfrac{1}{2}(1200)(20.^2 - 10.^2) = 600 \times 300 = 180{,}000\) J.

## Frequently asked questions

### Is energy ever destroyed?

No. Energy changes form but the total stays the same. When friction slows a sliding box, its kinetic energy becomes thermal energy (the surfaces warm up). When a book falls, potential energy becomes kinetic energy, then sound and heat when it lands.

### Why does the zero level for height not matter?

Only changes in potential energy affect the motion. If you measure height from the floor or from the table, PE values change, but the difference between two points stays the same. Pick whatever level makes the numbers simple, usually the lowest point.

### What is the difference between work and power?

Work is how much energy is transferred. Power is how fast it is transferred. Walking or running up the same stairs does the same work, but running does it in less time, so it takes more power.

## Sources

- [OpenStax College Physics 2e, 7.1 Work: The Scientific Definition](https://openstax.org/books/college-physics-2e/pages/7-1-work-the-scientific-definition), accessed 2026-10-01
- [OpenStax College Physics 2e, 7.2 Kinetic Energy and the Work-Energy Theorem](https://openstax.org/books/college-physics-2e/pages/7-2-kinetic-energy-and-the-work-energy-theorem), accessed 2026-10-01
- [OpenStax College Physics 2e, 7.3 Gravitational Potential Energy](https://openstax.org/books/college-physics-2e/pages/7-3-gravitational-potential-energy), accessed 2026-10-01
- [OpenStax College Physics 2e, 7.7 Power](https://openstax.org/books/college-physics-2e/pages/7-7-power), accessed 2026-10-01

## Related

- [Momentum and collisions](https://duckyhelper.com/learn/physics/momentum-and-collisions/)
- [The kinematics equations](https://duckyhelper.com/learn/physics/kinematics-equations/)
- [Friction: static and kinetic](https://duckyhelper.com/learn/physics/friction/)
- [Physics study guides](https://duckyhelper.com/learn/physics/)

## Try asking Ducky

- "I used conservation of energy on a problem with friction. Show me where it went wrong."
- "Check my angle on the work problem. Is it the angle to the ground or to the motion?"
- "Give me a roller coaster problem with two hills and let me try it."

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