Work and Power Calculator finds power from work and time and solves for any of the three. It also takes force and distance, force and velocity, a lifted mass, or a change in speed.
Power is work divided by time. Pick what to solve for and enter the other two values.
Result Unit
Calculation Steps
- Power: P = W ÷ t = 9,000 J ÷ 60 s = 150 W
Power by Time
| Time | Power | Horsepower |
|---|---|---|
| 15 s | 600 W | 0.8046 hp |
| 30 s | 300 W | 0.4023 hp |
| 60 s | 150 W | 0.2012 hp |
| 120 s | 75 W | 0.1006 hp |
| 240 s | 37.5 W | 0.05029 hp |
Each row does the same work in a different time. The highlighted row is your case.
What the Work and Power Calculator Solves
The Work and Power Calculator links three quantities, which are the work done, the time it takes and the power that results. With the loaded values of 9,000 J in 60 s, the power is 150 W, or 0.2012 mechanical horsepower. Pick Power, Work or Time under Solve For and the other two become the inputs.
The Calculate From list adds six more starting points for problems that do not hand you the work directly. Force, Distance and Time and Force and Velocity cover straight-line motion, and Torque and RPM covers a rotating shaft. Lifting a Mass, Climbing a Slope and Change in Speed deal with gravity and acceleration.
A work and power calculator for physics has to keep the units straight as well. Work can go in as joules, kilowatt-hours, foot-pounds, calories or BTU, and time in anything from milliseconds to days. Power comes out in watts, kilowatts, megawatts, both kinds of horsepower, foot-pounds per second or BTU per hour.
Power from Work and Time
Power is the rate of doing work. OpenStax University Physics, section 7.4 defines average power as the work done in a time interval divided by that interval. To calculate power based on work and time, divide the joules by the seconds.
$$P = \frac{W}{t}$$
P is power in watts, W is work in joules and t is time in seconds. One watt is one joule per second. Doing 9,000 J in 60 s works out to 150 J every second, so the power is 150 W. This is also how you convert work to power, since work on its own has no rate until a time is attached.
| Time for 9,000 J | Power | Horsepower |
|---|---|---|
| 15 s | 600 W | 0.8046 hp |
| 30 s | 300 W | 0.4023 hp |
| 60 s | 150 W | 0.2012 hp |
| 120 s | 75 W | 0.1006 hp |
| 240 s | 37.5 W | 0.05029 hp |
The table is the Power by Time table from the tool, and it shows the trade plainly. Halving the time doubles the power needed for the same work. A second card reads the result another way, as 15.3 kg lifted at a steady 1 m/s.
Power to work formula
Turn the formula around and work is power times time, which the Work and Power Calculator handles with Work under Solve For. A steady 250 W kept up for 60 s does 15,000 J of work, which is 4.167 Wh. Work grows in step with both the power and the time it runs.
$$W = P \times t$$
The same sum answers how much 10 minutes of electricity is. A 1,000 W appliance running for 10 minutes uses 600,000 J, which is 0.1667 kWh. To calculate power use over longer periods and its cost per unit, the Watt Converter has a kWh option with a price field.
Time from work and power
Used as a power, work and time calculator, the third option divides the work by the power. At 250 W, a job of 9,000 J takes 36 s. Doubling the power cuts that to 18 s, and one mechanical horsepower would finish it in 12.07 s. Long results switch to a readable unit, so 1 kWh of work at 100 W shows as 10 h.
$$t = \frac{W}{P}$$
Is Work Equal to Power?
No, and the lifting example shows the difference between power and work. Raising 60 kg through 3 m takes 1,765 J whether it is done fast or slowly. Done in 4 s it needs 441.3 W. Done in 16 s it needs only 110.3 W.
Work is the amount of energy transferred, measured in joules. Power is how fast that transfer happens, measured in watts. Two machines can do the same work, and the more powerful one simply finishes sooner.
The Three Formulas of Power
In mechanics the three formulas of power are work over time, force times distance over time, and force times velocity. They are the same idea written for different known values, and the Work and Power Calculator has an option for each.
$$P = \frac{W}{t} \qquad P = \frac{F \times d}{t} \qquad P = F \times v$$
Force, Distance and Time uses the work calculation formula W = F × d first. A 100 N force over 50 m does 5,000 J, and spread across 10 s that is 500 W. The same option can solve for the force, the distance or the time when the power is known.
Force and Velocity gives the power at one instant, with no distance or time needed. A 400 N force on an object moving at 25 m/s delivers 10,000 W, or 13.41 hp. OpenStax writes this as the force multiplied by the velocity of the body.
Both options take an angle between the force and the motion, and only the part of the force along the motion counts. At 60° the 500 W example drops to 250 W. Above 90° the power turns negative, because the force is taking energy away. The Work Calculator covers the angle itself, net work and friction in more depth.
Force and speed at fixed power
People looking for a mechanical power calculator often want this trade. For a fixed power, force and speed swap places, which is what a low gear does. The table holds the power at 5,000 W, about 6.7 hp.
| Speed | Force at 5,000 W | Pounds-force |
|---|---|---|
| 6.25 m/s | 800 N | 179.8 lbf |
| 12.5 m/s | 400 N | 89.92 lbf |
| 25 m/s | 200 N | 44.96 lbf |
| 50 m/s | 100 N | 22.48 lbf |
| 100 m/s | 50 N | 11.24 lbf |
Read the other way, 5,000 W against a steady 400 N of drag gives a top speed of 12.5 m/s, which is 45 km/h. Doubling the force to overcome halves that speed, and the same trade appears on a rotating shaft with torque in place of force.
Torque and RPM to power
On a shaft, power is torque times angular speed, which is equation 10.31 in OpenStax section 10.8. The Torque and RPM option converts the shaft speed to radians per second first and then multiplies.
$$P = \tau \times \omega \qquad \omega = \frac{2 \pi \times n}{60}$$
Here τ is the torque in newton-meters, ω is the angular speed in rad/s and n is the speed in rpm. A torque of 200 N·m at 3,000 rpm is 62,830 W, or 84.26 hp. In US units the rule is hp = lbf·ft × rpm ÷ 5,252, so 300 lbf·ft at 5,252 rpm is exactly 300 hp.
The option also solves for torque or shaft speed. A shaft carrying 50,000 W at 3,000 rpm turns with 159.2 N·m, and against a 200 N·m load it would run at 2,387 rpm. OpenStax works a boat engine of 90,000 W at 300 rpm and finds about 2,865 N·m on the propeller shaft, the same figure the tool returns.
Power to Lift a Mass
The Lifting a Mass option is the work calculator with mass and distance that many homework problems need. The work is the weight times the height, and the power is that work over the time taken.
$$P = \frac{m \times g \times h}{t}$$
Here m is the mass, g is gravity and h is the height gained. The loaded case is 60 kg raised 3 m in 4 s, which takes 1,765 J and 441.3 W. With 500 W supplied instead, the same lift takes 3.53 s. That power could raise 67.98 kg in the 4 s, or lift the 60 kg to 3.399 m.
OpenStax works a pull-up in which 72 kg of body mass rises 0.60 m in 0.8 s and gets about 529 W. The Work and Power Calculator gives 529.6 W for the same numbers. The small gap comes from the book rounding gravity to 9.8 m/s², while the gravity field loads 9.80665 m/s².
The same lift formula is behind the Potential Energy Calculator, since m × g × h is the energy stored by the height. Here the time field turns that energy into a power rating.
Power to climb a slope
Climbing a Slope takes a mass, a speed and a grade in percent or degrees. Only the part of the weight along the slope has to be overcome, so the power is that force times the speed.
$$P = m \times g \times v \times \sin\theta$$
A 1,500 kg car holding 20 m/s up an 8% grade needs 23,460 W, or 31.46 hp, and gains 1.595 m of height every second. That figure covers gravity alone, and rolling resistance and air drag come on top. The option can also find the speed or the mass a given power will carry uphill.
Motor power with efficiency
A real hoist, pump or engine draws more than it delivers. Every option except Work and Time has an Efficiency field under the options bar. Set it below 100% and the 441.3 W lift needs 551.6 W of input at 80% efficiency, 490.3 W at 90% and 735.5 W at 60%.
OpenStax takes a 1,200 kg car climbing a 15% grade at 25 m/s, with 75% of the engine power going against gravity. Climbing a Slope gives 43,640 W for the climb itself. With Efficiency at 75% the input power is 58,190 W, which matches the book’s 58 kW.
Average and Peak Power in Acceleration
In the Work and Power Calculator, Change in Speed finds the work from kinetic energy and divides it by the time. A 1,200 kg car going from rest to 27 m/s gains 437,400 J. Over 8 s that is an average of 54,680 W, or 73.32 hp.
$$P = \frac{m \times (v_1^2 – v_0^2)}{2 \times t}$$
Average power hides the peak. With a steady force of 4,050 N, power rises with speed and reaches 109,400 W at the end, twice the average. The car covers 108 m meanwhile and works at 45.56 W for each kilogram.
Solving for time, 60,000 W of average power gets the same car to 27 m/s in 7.29 s. Swap the two speeds and the power turns negative, which is the rate at which brakes must remove energy. The Kinetic Energy Calculator gives the energy side on its own, and the Power-to-Weight Ratio Calculator compares vehicles by power per unit of weight.
Watts, Horsepower and Joules
One unit of power, the watt, equals one joule per second. A kilowatt is 1,000 W, and a kilojoule is 1,000 J. One kilowatt running for an hour does 1 kWh of work, which is 3.6 MJ.
Horsepower comes in two common sizes. Mechanical horsepower is 745.7 W, or 550 foot-pounds per second, and metric horsepower is 735.5 W in NIST Special Publication 811. Many pages quote 735.5 W as “1 horsepower” without saying it is the metric one. The Work and Power Calculator shows both on every power card.
How to calculate 3 phase AC power?
Electrical power follows a different set of formulas. The three formulas of electricity are P = V × I, P = I² × R and P = V² ÷ R. Three-phase AC power is the square root of 3 times the line voltage, the current and the power factor. A 400 V supply at 20 A and a power factor of 0.85 carries 11.78 kW, and the Watt Converter works these cases out.
Powers in Math: 2 to the Power of n
Some searches for a power calculator mean exponents, not watts. In math, a power is a number multiplied by itself a set number of times, so 7 to the power of 2 is 7 × 7 = 49. This tool works on physical power, so the common powers of 2 are listed here.
| Power of 2 | Value |
|---|---|
| 2 to the power of 7 | 128 |
| 2 to the power of 8 | 256 |
| 2 to the power of 9 | 512 |
| 2 to the power of 10 | 1,024 |
| 2 to the power of 16 | 65,536 |
| 2 to the power of 20 | 1,048,576 |
| 2 to the power of 50 | 1,125,899,906,842,624 |
| 2 to the power of 100 | about 1.268 × 10³⁰ |
| 2 to the power of 2000 | about 1.148 × 10⁶⁰², a number with 603 digits |
The simple way to find powers of 2 is to keep doubling, or to square a known one. Since 2 to the power of 10 is 1,024, squaring it gives 2 to the power of 20. For a rough size of 2 to the power of 100, treat each 2¹⁰ as about 1,000 and the answer is near 10³⁰. On a calculator, use the key marked xy or ^, as in 2 ^ 8 = 256.
Half of 2 to the power of 10 is 2 to the power of 9, which is 512, because halving removes one factor of 2. The number 128 is itself a power of 2, namely 2 to the power of 7. Any number to the power of 1 is itself, and 1 to any power is still 1.
The math that says “I love you” is the inequality 9x − 7i > 3(3x − 7u). Expanding the right side gives 9x − 21u. Taking 9x from both sides and dividing by −7 flips the sign and leaves i < 3u, which reads as a heart between “i” and “u”.