Power is calculated by dividing work (or energy) by time: Power = Work ÷ Time. In SI units, power is measured in watts, where one watt equals one joule of energy transferred per second. That core formula applies whether you’re solving a physics problem, sizing a motor, or estimating athletic output, though each field has its own practical variations worth knowing.
The Basic Formula: Work Divided by Time
The fundamental equation is straightforward. If you push a box across a floor, doing 200 joules of work in 10 seconds, your power output is 20 watts. The formula works for any situation where energy is transferred over a period of time.
There’s an equally useful alternative form: Power = Force × Velocity. This version is handy when you know how much force is being applied to something already in motion. A cyclist pushing 50 newtons of force while traveling at 8 meters per second produces 400 watts of power. Both formulas give the same result; they’re just rearranged to fit different starting information.
Units of Power
The SI unit is the watt (W), which breaks down to 1 kilogram × meter² per second³. For larger systems like engines and motors, power is often expressed in horsepower. One imperial horsepower equals roughly 745.7 watts, defined originally as 550 foot-pounds of force per second. Metric horsepower (often labeled PS or cv) is slightly smaller at about 735.5 watts. To convert, just multiply horsepower by the appropriate factor.
Power in Rotating Systems
Engines, turbines, and electric motors all involve rotation, so the formula takes a slightly different shape: Power = Torque × Angular Velocity. Torque is the rotational force (measured in newton-meters), and angular velocity is the speed of rotation measured in radians per second. If you’re working with RPM instead, convert first: multiply RPM by 2π and divide by 60 to get radians per second. A motor producing 50 newton-meters of torque at 300 RPM (31.4 rad/s) delivers about 1,570 watts.
This same principle is how cycling power meters work. They measure torque at the pedal, crank, or hub, then multiply it by cadence to calculate the rider’s power output in watts.
Electrical Power Calculations
For direct current (DC) circuits, electrical power is simply voltage multiplied by current: P = V × I. A 12-volt battery delivering 3 amps produces 36 watts.
Alternating current (AC) circuits are more complex because voltage and current don’t always peak at the same moment. This timing mismatch, called the phase angle, splits power into three types:
- Real power is the energy actually consumed and converted to useful work, measured in watts. It’s calculated as P = V × I × cos(θ), where θ is the phase angle between voltage and current.
- Reactive power is the energy that sloshes back and forth in the circuit without doing useful work, measured in volt-amperes reactive (VAR). It’s calculated as Q = V × I × sin(θ).
- Apparent power is the total power the source must deliver to keep the circuit running, measured in volt-amperes (VA). It’s simply S = V × I, and it relates to the other two by S² = P² + Q².
The ratio of real power to apparent power is the power factor, equal to cos(θ). A power factor of 0.87 means 87% of the supplied current performs real work, while the remaining 13% is reactive. Higher power factor means a more efficient system. Industrial facilities with lots of motors often have low power factors and install correction equipment to bring them closer to 1.0.
Estimating Athletic Power Output
In sports science, power measures how explosively an athlete can produce force. The simplest field test is the vertical jump. A widely used formula developed by Sayers and colleagues estimates peak power from just two measurements:
Peak Power (watts) = 60.7 × jump height (cm) + 45.3 × body mass (kg) − 2,055
So a 75 kg athlete with a 55 cm vertical jump would produce an estimated peak power of about 1,733 watts. This gives coaches a quick way to track explosive strength without expensive lab equipment.
Statistical Power in Research
If you landed here from a statistics or research methods class, “power” means something entirely different. Statistical power is the probability that a study will correctly detect a real effect when one exists. It’s calculated as 1 minus the probability of a Type II error (a false negative, meaning you miss a real finding).
Four factors determine statistical power, and they’re all connected:
- Sample size: Larger samples give more power. This is the factor researchers have the most control over.
- Effect size: The bigger the real difference or relationship you’re trying to detect, the easier it is to find, and the more power you have. Small effects require much larger samples.
- Significance level (alpha): This is the threshold for declaring a result “real,” typically set at 0.05. A more lenient alpha increases power but also raises the risk of a false positive.
- Variability: More noise in the data makes it harder to spot a signal, reducing power.
The standard target for power in most research is 0.80, meaning an 80% chance of detecting a true effect. Researchers run what’s called a power analysis before collecting data, plugging in their expected effect size and chosen alpha to determine how many participants they need. When sample size stays constant, power drops as the expected effect size gets smaller, which is why studies looking for subtle effects need substantially more participants.