Mechanical Power

Physics Classical Mechanics • Work Energy and Power

Written by STEM Calculators Team Published May 9, 2026 Updated May 9, 2026

Calculate average mechanical power from work over time, instantaneous power from \(P=\vec F\cdot\vec v\), or a variable force-speed profile using numerical averaging.

Preset

Power model

Solve for

Power output

Instantaneous power inputs

\(P_{\mathrm{inst}}=Fv\cos\varphi\), where \(\varphi\) is the angle between the force and velocity vectors. Positive power adds mechanical energy; negative power removes it.

Force magnitude F

Speed v

Angle φ between F and v

Instantaneous power P

Average power inputs

The calculator first computes work \(W=Fs\cos\alpha\), then uses \(P_{\mathrm{avg}}=W/\Delta t\).

Force magnitude F

Displacement s

Angle α between F and s

Time interval Δt

Average power P_avg

Variable force-speed profile

Force, speed, and angle are linearly interpolated from start to end. The calculator estimates \(\overline P=\frac{1}{T}\int_0^T F(t)v(t)\cos\varphi(t)\,dt\) by trapezoidal integration.

Start force F₀

End force F₁

Start speed v₀

End speed v₁

Start angle φ₀

End angle φ₁

Total time T

Work output

Speed output

Diagram zoom

Animation speed

Instantaneous power: \[ P=\vec F\cdot\vec v=Fv\cos\varphi. \] Average power: \[ P_{\mathrm{avg}}=\frac{W}{\Delta t},\qquad W=Fs\cos\alpha. \]

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Enter the force, speed, angle, work, or time values, then click “Calculate”.

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Theory – Mechanical power

Mechanical power measures how quickly mechanical work is done or how quickly mechanical energy is transferred. The SI unit of power is the watt:

\[ 1\ \mathrm{W}=1\ \mathrm{J\,s^{-1}}. \]

1. Average power

Average power over a finite time interval is

\[ P_{\mathrm{avg}}=\frac{W}{\Delta t}. \]

If a constant force moves an object through a displacement \(s\), the work done by that force is

\[ W=Fs\cos\alpha, \]

where \(\alpha\) is the angle between the force vector and the displacement vector. Therefore,

\[ P_{\mathrm{avg}} = \frac{Fs\cos\alpha}{\Delta t}. \]

If the force points in the same direction as the displacement, then \(\alpha=0^\circ\) and the work is positive. If the force points opposite the displacement, then \(\alpha=180^\circ\) and the work is negative. If the force is perpendicular to the displacement, then \(\alpha=90^\circ\) and the work is zero.

2. Instantaneous power

Instantaneous power describes the rate of energy transfer at one particular moment. In vector form,

\[ P=\vec F\cdot\vec v. \]

Using the dot product,

\[ P=Fv\cos\varphi, \]

where \(\varphi\) is the angle between the force and velocity vectors. Only the component of force parallel to the velocity transfers mechanical energy:

\[ F_\parallel=F\cos\varphi, \qquad P=F_\parallel v. \]

3. Sign of power

Angle range	Sign of \(P\)	Physical meaning
\(0^\circ\le\varphi<90^\circ\)	Positive	The force adds mechanical energy to the object.
\(\varphi=90^\circ\)	Zero	The force does no work at that instant.
\(90^\circ<\varphi\le180^\circ\)	Negative	The force removes mechanical energy from the object.

4. Variable force and speed

If force, speed, or angle changes with time, power is also time-dependent:

\[ P(t)=F(t)v(t)\cos\varphi(t). \]

The total work transferred over a time interval is

\[ W=\int_0^T P(t)\,dt. \]

The average power over that interval is

\[ \overline P=\frac{1}{T}\int_0^T P(t)\,dt. \]

The calculator’s variable-profile mode linearly interpolates the force, speed, and angle between start and end values, then estimates the integral numerically.

5. Worked example

A force of \(800\ \mathrm{N}\) pushes an object moving at \(12\ \mathrm{m/s}\). The force makes an angle of \(25^\circ\) with the velocity. The instantaneous power is

\[ P=Fv\cos\varphi. \]

Substitute:

\[ P=(800)(12)\cos(25^\circ). \]

Since \(\cos(25^\circ)\approx0.906\),

\[ P\approx9600(0.906)\approx8700\ \mathrm{W}. \]

So the force transfers energy at a rate of about

\[ \boxed{8.7\ \mathrm{kW}}. \]

Formula summary

Concept	Formula	Use
Average power	\(P_{\mathrm{avg}}=\frac{W}{\Delta t}\)	Power over a finite time interval.
Work by constant force	\(W=Fs\cos\alpha\)	Work from force, displacement, and angle.
Instantaneous power	\(P=\vec F\cdot\vec v\)	Power at one instant.
Dot product form	\(P=Fv\cos\varphi\)	Power from force magnitude, speed, and angle.
Parallel force component	\(F_\parallel=F\cos\varphi\)	Only this component changes mechanical energy.
Variable-profile work	\(W=\int_0^T P(t)\,dt\)	Total work when force or speed varies.

Mechanical power is not just force times speed. It is force component along motion times speed: \(P=F_\parallel v\).

Frequently Asked Questions

What is mechanical power?

Mechanical power is the rate at which mechanical work is done or energy is transferred. Its SI unit is the watt, where 1 W = 1 J/s.

What is the formula for average mechanical power?

Average mechanical power is P_avg = W / delta t. If the work comes from a constant angled force, W = F s cos(alpha).

What is the formula for instantaneous mechanical power?

Instantaneous mechanical power is P = F dot v = F v cos(phi), where phi is the angle between force and velocity.

Why does the angle matter in mechanical power?

Only the force component parallel to motion transfers mechanical energy. That component is F_parallel = F cos(phi).

When is mechanical power positive?

Power is positive when the force has a component in the same direction as velocity or displacement.

When is mechanical power zero?

Power is zero when the force is perpendicular to velocity or displacement, so cos(90 degrees) = 0.

When is mechanical power negative?

Power is negative when the force has a component opposite the motion, such as braking or drag.

How does the calculator handle variable force and speed?

In profile mode, the calculator linearly interpolates force, speed, and angle, computes P(t) = F(t)v(t)cos(phi(t)), and estimates total work using trapezoidal integration.

What units should I use for input?

Use newtons for force, meters for displacement, seconds for time, meters per second for speed, degrees for angles, joules for work, and watts for power.

What is the result for 800 N at 25 degrees and 12 m/s?

The instantaneous power is P = 800 × 12 × cos(25 degrees), which is approximately 8700 W.

Mechanical Power

Instantaneous power inputs

Average power inputs

Variable force-speed profile