Instantaneous Velocity and Speed

Measuring Speed over a Shrinking Time Interval

Instantaneous speed is the speed of an object at a particular time or at a particular point on its path. A common example is the number shown by a vehicle speedometer when the driver looks at it.

Instantaneous velocity is also measured at a particular time, but it includes direction. If the vehicle is moving right, the velocity is positive. If it is moving left, the velocity is negative.

An instantaneous value answers, "How fast right now?" An average value covers a longer part of the trip.

\begin{aligned} &\text{instantaneous speed}=\lvert\vec{v}_{\text{instant}}\rvert \end{aligned}

Instantaneous speed is the magnitude of instantaneous velocity.

If position is a differentiable function of time, instantaneous velocity is the limit of average velocity as the sampling interval shrinks to zero.

v_x(t)=\lim_{\Delta t\to 0}\frac{\Delta x}{\Delta t}=\frac{dx}{dt}

Real instruments estimate this value over a short but finite sampling interval. A shorter interval can better approximate the value at that instant when the motion varies smoothly.

Measurement at One Point

Choose a measurement time and compare the car's instantaneous speed and velocity as it passes the measuring point.

Measurement time: $t=6\text{ s}$
Instantaneous speed: $20\text{ m/s}$
Instantaneous velocity: $+20\text{ m/s}$

Direction Sets the Sign of Instantaneous Velocity

The measuring point in the simulation records the car at one instant. Speed gives the magnitude of its motion. The direction of travel sets the sign of its velocity.

At $t=6 \text{ s}$ , the car moves rightward with an instantaneous speed of $20 \text{ m/s}$ . Because rightward is chosen as the positive direction, the instantaneous velocity is $+20 \text{ m/s}$ .

At $t=9 \text{ s}$ , the car moves leftward with an instantaneous speed of $8 \text{ m/s}$ . Because leftward is opposite the positive direction, the instantaneous velocity is $-8 \text{ m/s}$ .

Time	Speed reading	Direction of motion	Instantaneous velocity
$6 \text{ s}$	$20 \text{ m/s}$	rightward	$+20 \text{ m/s}$
$9 \text{ s}$	$8 \text{ m/s}$	leftward	$-8 \text{ m/s}$

Instantaneous speed has no direction sign. Instantaneous velocity uses a positive or negative sign to show direction along the chosen axis.

Difference from Average Speed

Average speed summarizes the whole trip by dividing total distance by total time. Instantaneous speed gives the speed at one chosen time.

If a speedometer shows $15 \text{ m/s}$ right now, it does not mean the vehicle moved at $15 \text{ m/s}$ for the whole trip.

For example, a vehicle may stop at a red light and then move quickly afterward. Its average speed can be small because the whole trip time is counted, while its instantaneous speed while moving can be much larger.

A Speedometer Measures the Current Speed

A posted limit of $80 \text{ km/h}$ applies to the vehicle at every moment. A roadside detector therefore reads the instantaneous speed as the vehicle passes.

A detector reads the speed at the moment the vehicle passes it, so it needs no destination.

Here, "instantaneous" means that the measurement applies to one particular time.