In everyday life, work means effort. In physics, work is a force moving an object over a distance. When a force pushes or pulls something along its direction of motion, the force transfers energy to the object.
Force is measured in newtons (N) and distance in meters (m), so work is measured in newton-meters. We call one newton-meter a joule (J).
For a force to do work on an object, three things have to be true:
The same goes for carrying a backpack across a room at a steady height. Your arms push up, the backpack moves sideways, and a force at 90° to the motion does no work.
Often a force doesn’t point exactly along the motion. If you pull a sled with a rope angled upward, only part of your pull drags the sled forward. The rest lifts it slightly. We handle that with the angle θ between the force and the direction of motion:
Think of cosθ as the fraction of the force that points along the motion.
| Angle θ | cosθ | What it means |
|---|---|---|
| 0° | 1.00 | Force points straight along the motion |
| 30° | 0.87 | Most of the force helps |
| 45° | 0.71 | About 70% of the force helps |
| 60° | 0.50 | Half of the force helps |
| 90° | 0.00 | Force is straight across the motion: none of it helps |
| 180° | −1.00 | Force points straight against the motion |
The cosine goes negative when the angle is bigger than 90°. That’s how the math tells you a force is working against the motion. Work is a quantity with a sign, and the sign tells you what happens to the object’s speed.
| Situation | Angle | Work | Effect |
|---|---|---|---|
| Force points with the motion | 0° | Positive (+) | Speeds the object up |
| Force points against the motion | 180° | Negative (−) | Slows the object down |
| Force is straight across the motion | 90° | Zero (0) | Speed doesn’t change (direction can) |
| Object doesn’t move | any | Zero (0) | Nothing to change |
Play with the three variables. Set the angle to 90° and see what happens. Then push it past 90° and watch the sign flip.
The block always moves to the right. Change the force, the distance, and the angle of the force, and watch the sign of the work.
Kinetic energy (KE) is the energy an object has because it is moving. It depends on how much mass is moving and, much more strongly, on how fast:
| Object | Mass | Speed | KE |
|---|---|---|---|
| Lab cart | 2 kg | 3 m/s | 9 J |
| Person jogging | 60 kg | 3 m/s | 270 J |
| Same person, twice as fast | 60 kg | 6 m/s | 1,080 J |
| Car on a road | 1,000 kg | 20 m/s | 200,000 J |
You’ll go deeper on kinetic energy in Activity 4-1-2. For now, all you need is that a moving object carries energy, and the formula above tells you how much.
Here is the big idea. Add up the work done by every force on an object, and you get the change in the object’s kinetic energy:
In plain words: net positive work speeds an object up, net negative work slows it down, and net zero work leaves its speed unchanged. The theorem lets you skip the messy details of the motion. You just count the work.
Count every force.For the block in the Slide & Stop lab, here are the forces and what each one does:
| Force | Angle to motion | Work |
|---|---|---|
| Gravity (pulls down) | 90° to the slide | 0 |
| Floor’s normal force (pushes up) | 90° to the slide | 0 |
| Friction (points backward) | 180° to the slide | Negative |
Only friction does work, so friction’s work is the net work. That’s why measuring friction and stopping distance tells you the block’s starting energy.
The theorem gives a stopping-distance rule: d = ½ m v² ÷ F_f. Change the sliders and watch how strongly speed controls the distance.
A block slides to a stop with only friction acting. Use the theorem to find how far it goes: friction’s work must erase all the starting kinetic energy, so F_f × d = ½ m v².
Four questions. They are not graded. Answer them to find out what still needs work.
Work and the work-energy theorem show up in both of the big builds that this activity sets up: