Instantaneous Velocity and Speed

A Reading at One Instant

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 read at a particular time, but it still 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?", not "how fast since the trip began?"

\text{instantaneous speed}=|\vec{v}_{\text{instant}}|

Instantaneous speed is the magnitude of instantaneous velocity.

If motion is recorded as position data over time, the instantaneous value can be imagined as a position change over a very small time interval around the instant being read.

v_{\text{instant}}\approx \frac{\Delta x}{\Delta t}\quad \text{for very small } \Delta t

The smaller the time interval, the closer the reading is to the value at that exact instant. That is why an instantaneous value does not need to wait until the whole trip is over.

Reading at a Measuring Point

Choose a reading moment to see the car pass a measuring point and its instantaneous velocity value.

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

The measuring point in the simulation represents the place where motion is read at one instant. The speed gives the magnitude of the reading, while the car's direction as it passes the measuring point 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	$-8 \text{ m/s}$

So instantaneous speed is always written as an unsigned magnitude, while instantaneous velocity uses a sign to show direction.

Not the Trip Average

Instantaneous speed is different from average speed. Average speed uses the whole distance and the whole time, while instantaneous speed reads the state at one 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.

Speedometer Reads One Moment

Speed-limit signs and road speed detectors read an instantaneous state. When a limit says $80 \text{ km/h}$ , it means a vehicle must not exceed that speed at the moment it passes.

That is why a detector does not need to wait for the vehicle to reach its destination. It checks the speed at the moment the vehicle passes the measuring point.

So, the word "instantaneous" emphasizes that the motion value is read at one particular time, not across the whole trip.