What this calculator does
Find out how much energy a push or pull puts into moving something, and how fast it does it. In physics, that energy is called work. Only the part of the force that points along the motion counts. So a force at an angle θ (theta) to the direction of travel does work W = F × d × cos θ. Here F is the force, d is the distance, and cos θ is the share of the force that points along the path.
Pushing a box straight along the floor uses all of the force. Pulling a sled with a rope angled upward uses only the part that points forward. Carrying a bag level across a room does no work on the bag at all. That's because the lifting force is at 90° to the motion.
Power is how quickly that work is done: P = W ÷ t, where t is the time. Two people who lift the same load up the same stairs do the same work. But the one who climbs faster delivers more power.
This calculator handles both equations. It solves for work, force, distance, angle, power or time. It accepts joules, kilojoules, foot-pounds-force, watt-hours and kilowatt-hours. It reports power in watts, kilowatts, mechanical horsepower or foot-pounds-force per second.
Where it's used
Why this math matters outside the classroom:
- Pulling a wagon. A child pulling a wagon 20 m with 50 N at 30° to the ground does 50 × 20 × cos 30° ≈ 866 J of work.
- Stair power. A 70 kg student who climbs 4 m of stairs in 5 s puts out about 549 W, roughly 0.74 horsepower.
- Hoist motors. Engineers size a hoist motor from the work needed to lift a load and how quickly the lift must happen.
- Carrying a box. Physics students learn that carrying a box level across a room does no work against gravity, since force and motion are at 90°.
Grade level and standards: Grades 10–11; Common Core State Standards for Mathematics HSG.SRT.C.8, HSA.CED.A.4. © Copyright 2010. National Governors Association Center for Best Practices and Council of Chief State School Officers. All rights reserved.
How to use it
- Choose what to solve for. The first four options use W = F × d × cos θ. Power and time use P = W ÷ t.
- Enter the known values with their units. The angle is measured between the force and the direction of motion.
- When finding work, you can also add the time taken to get the average power.
- Pick the output unit and read the answer, which is also listed in the other units.
What the inputs mean
- Force (F)
- The size of the force. Its direction relative to the motion is set by the angle.
- Angle between force and motion (θ)
- Use 0° when the force points exactly along the motion. Between 90° and 180°, the force works against the motion, and the work is negative. Friction and braking work that way.
- Work (W)
- Work and energy share units. A negative value means energy was taken out of the object rather than put in.
- Power (P)
- Mechanical horsepower is set at 550 foot-pounds-force per second, which is 745.69987158227022 W. Metric horsepower and electrical horsepower are slightly different units and aren't used here.
- Time taken (t)
- The time over which the work was done. Power from work and time is an average. The power at any single moment can be higher or lower.
Formula and method
W = F × d × cos θ F = W ÷ (d × cos θ) d = W ÷ (F × cos θ) θ = arccos(W ÷ (F × d)) P = W ÷ t t = W ÷ P 1 hp = 550 ft·lbf/s = 745.69987158227022 W
W is work, F is force, d is distance and θ is the angle between the force and the motion. P is power and t is time. arccos is the inverse of cosine: it turns a cosine back into an angle. ft·lbf/s means foot-pounds-force per second.
The calculator first converts force and distance to newtons and meters. So work comes out in joules (1 J = 1 N·m). The cosine shrinks the force down to the part that lies along the path: cos 0° = 1, cos 60° = 0.5 and cos 90° = 0. Exact quarter turns are handled exactly, so 90° gives zero work with no rounding noise.
Power divides the work, in joules, by the time in seconds to give watts (1 W = 1 J/s). When solving backwards, the calculator refuses combinations that no real angle allows. One example is more work than force × distance. Another is positive work with an angle above 90°.
Worked example
You pull a loaded sled 10 m with a rope that makes a 30° angle with the ground, using a steady 50 N pull, and it takes 5 seconds.
- cos 30° ≈ 0.8660254.
- W = 50 × 10 × 0.8660254 ≈ 433.0127 J.
- P = 433.0127 ÷ 5 ≈ 86.6025 W.
- In horsepower: 86.6025 ÷ 745.69987 ≈ 0.1161 hp.
Result: The pull does about 433.01 J of work on the sled, at an average power of about 86.60 W (0.116 hp).
Understanding the result
Positive work speeds an object up or raises its energy. Negative work slows it down. Zero work at 90° explains why a satellite in a circular orbit keeps its speed. Gravity always pulls at right angles to its motion.
For a sense of scale, 1 kWh is 3.6 million joules. A fit person can keep up very roughly 100 W of mechanical output for a long stretch. Short sprints can reach several times that.
Assumptions and limitations
- The force is constant in size and direction over the distance.
- The path is a straight line, so a single angle describes the force.
- Power is averaged over the whole time.
- A force that changes along the path needs calculus (the work integral, W = ∫F·ds) instead of this formula. It adds up the work over each tiny piece of the path.
- The result is the work done by this one force. It isn't the net work of all the forces acting on the object.
- Energy lost to friction or heat inside a machine isn't included. So a real motor or engine will need more power going in than the output found here.
Frequently asked questions
Why is no work done when I hold a heavy object still?
Work needs movement in the direction of the force. Holding a weight still involves no movement, so W = 0 in the physics sense. Your muscles still use chemical energy to stay tensed, though.
Can work be negative?
Yes. When the angle between the force and the motion is greater than 90°, cos θ is negative, and so is the work. Friction on a sliding box and the brakes on a moving car both do negative work. They remove kinetic energy (energy of motion).
What is the difference between work and power?
Work is an amount of energy transferred, measured in joules. Power is the rate of that transfer, measured in watts. Doing the same work in half the time takes twice the power.
Which horsepower does this use?
Mechanical (imperial) horsepower: 550 ft·lbf per second, about 745.7 W. Metric horsepower is about 735.5 W. So the two differ by roughly 1.4%.
How does work relate to kinetic energy?
The net work done on an object equals its change in kinetic energy (energy of motion). That's called the work–energy theorem. Say 433 J of net work is done on a sled that starts at rest. It ends up with 433 J of kinetic energy, ignoring losses.
How this calculator is checked
Method: Definition of mechanical work for a constant force (W = F·d·cos θ) and of average power (P = W/t); horsepower = 550 ft·lbf/s by definition. The calculation is covered by automated tests with known reference values and edge cases. Last updated 2026-10-18. Found a problem? Tell us.
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