KAHIBARO
Table of Contents
Power as a Rate
In mechanics, power tells us how fast work is being done or how fast energy is being transferred. Average power describes this over a time interval, but instantaneous power describes it at one exact moment.
If a force acts on an object and the object moves, the power at that instant depends on both the force and the velocity at that same instant.
Instantaneous power is the time rate of change of work:
$$
P = \frac{dW}{dt}
$$
Its SI unit is the watt, where
$$
1\ \text{W} = 1\ \text{J/s}
$$
From Work to Instantaneous Power
For a very small displacement $d\vec r$, the small amount of work done by a force $\vec F$ is
$$
dW = \vec F \cdot d\vec r
$$
Divide both sides by $dt$:
$$
\frac{dW}{dt} = \vec F \cdot \frac{d\vec r}{dt}
$$
Since $\frac{d\vec r}{dt} = \vec v$, we get the main formula for instantaneous power:
The instantaneous power delivered by a force is
$$
P = \vec F \cdot \vec v
$$
This can also be written as
$$
P = Fv\cos\theta
$$
where $\theta$ is the angle between the force and the velocity.
This formula is very important because it shows that only the component of force in the direction of motion contributes to power.
Meaning of the Dot Product in Power
The dot product makes the physical meaning clear. A force can have different directions relative to the motion.
| Direction of force relative to velocity | Value of $P$ | Meaning |
|---|---|---|
| Same direction | Positive | Force transfers energy to the object |
| Opposite direction | Negative | Force removes energy from the object |
| Perpendicular | Zero | Force does no work at that instant |
If the force points partly along the motion, only that parallel part matters for power.
If $\vec F \perp \vec v$, then
$$
P = \vec F \cdot \vec v = 0
$$
So a perpendicular force does zero instantaneous power.
Positive, Negative, and Zero Power
Positive power means energy is being given to the object. For example, an engine pushing a car forward while the car moves forward does positive power.
Negative power means energy is being taken away from the object. Friction often does negative power because it acts opposite the motion.
Zero power means no energy is transferred by that force at that instant. In uniform circular motion, the centripetal force points toward the center while the velocity is tangent to the circle, so they are perpendicular. Therefore the centripetal force does no instantaneous power.
Instantaneous Power and Kinetic Energy
Using the work energy theorem, power can also be connected to kinetic energy. Since work changes kinetic energy,
$$
dW = dK
$$
so
$$
P = \frac{dW}{dt} = \frac{dK}{dt}
$$
This means instantaneous power tells us how quickly kinetic energy is changing.
For a particle, net instantaneous power equals the rate of change of kinetic energy:
$$
P_{\text{net}} = \frac{dK}{dt}
$$
Special Case of Motion in One Dimension
In one dimensional motion, force and velocity lie along the same line, so the formula becomes especially simple:
$$
P = Fv
$$
Here the sign matters. If $F$ and $v$ have the same sign, power is positive. If they have opposite signs, power is negative.
For example, if a force of $10\ \text{N}$ acts on an object moving at $3\ \text{m/s}$ in the same direction, then
$$
P = 10 \times 3 = 30\ \text{W}
$$
If the same force acts opposite to the motion, then
$$
P = -30\ \text{W}
$$
Example of Instantaneous Power
Suppose a cyclist moves with instantaneous velocity
$$
\vec v = 6\,\hat{i}\ \text{m/s}
$$
and the force from the ground on the cyclist is
$$
\vec F = 20\,\hat{i}\ \text{N}
$$
Then
$$
P = \vec F \cdot \vec v = (20)(6) = 120\ \text{W}
$$
So the force delivers energy at a rate of $120\ \text{J/s}$.
Now suppose instead that the force were
$$
\vec F = 20\,\hat{j}\ \text{N}
$$
while the velocity stayed along $\hat{i}$. Then
$$
P = \vec F \cdot \vec v = 0
$$
because the force is perpendicular to the motion.
Visualizing Instantaneous Power
The first picture shows a force making an angle with the velocity, so only part of the force contributes to power. The second shows a perpendicular force, giving zero power. The third shows a force opposite the velocity, giving negative power.
Practical Interpretation
Instantaneous power is useful when force or speed changes from moment to moment. A car climbing a hill, a runner accelerating, or a machine lifting a load may not operate at a constant rate. Instantaneous power tells us the energy transfer at the exact current moment.
For a constant force, power can still vary if velocity varies. Likewise, for constant velocity, power can vary if force varies.
Instantaneous power depends on the present values of force and velocity, not on their past average values:
$$
P(t) = \vec F(t) \cdot \vec v(t)
$$
Summary Formula Table
| Quantity | Formula | Meaning |
|---|---|---|
| Instantaneous power | $P = \dfrac{dW}{dt}$ | Rate of doing work at one instant |
| Force and velocity form | $P = \vec F \cdot \vec v$ | Power delivered by a force |
| Magnitude form | $P = Fv\cos\theta$ | Depends on angle between force and motion |
| Net power | $P_{\text{net}} = \dfrac{dK}{dt}$ | Rate of change of kinetic energy |
Instantaneous power is the exact rate of energy transfer at a given moment. In mechanics, it is found most directly from the dot product of force and velocity.