Instantaneous Velocity Formula: v = dx/dt with Examples

The instantaneous velocity formula is $v = \frac{dx}{dt}$ , the limit of $\frac{\Delta x}{\Delta t}$ as the time interval shrinks toward zero. Average velocity looks at a whole stretch of motion. Instantaneous velocity zooms in until the stretch is a single moment and asks how fast, and in which direction, the object is moving right then.

In everyday terms it is what a speedometer with a direction arrow would read. In Class 11 it is introduced with calculus, and in AP Physics 1 it is read from the slope of a tangent on a position time graph. Both methods give the same number, and this page shows each one.

Quick facts

Formula
$v = \frac{dx}{dt}$
SI unit
m/s
Dimensional formula
[M⁰ L¹ T⁻¹]
Type
Vector, along the direction of motion at that instant
Graph
Slope of the tangent to the position time curve
Its size
Instantaneous speed, the speedometer reading

Instantaneous velocity formula physics: the idea of a limit

Take a stone falling so that its position is $x = 5t^2$ metres. To find its velocity at t = 3 s, work out the average velocity over shorter and shorter intervals starting at 3 s:

Average velocity over shrinking intervals from t = 3 s
Interval Δt	Δx	Δx / Δt
1 s	80 − 45 = 35 m	35 m/s
0.1 s	48.05 − 45 = 3.05 m	30.5 m/s
0.01 s	45.3005 − 45 = 0.3005 m	30.05 m/s
0.001 s	0.030005 m	30.005 m/s

The averages close in on 30 m/s. That limiting value is the instantaneous velocity. Calculus does the same job in one line: $\frac{d}{dt}(5t^2) = 10t$ , and at t = 3 s this gives 30 m/s.

Formula for instantaneous velocity: solved examples

Example 1: a polynomial position

A bead slides along a wire so that $x = 4t^3 - 2t + 7$ (SI units throughout). What is its velocity when t = 2 s?

$v = \frac{dx}{dt} = 12t^2 - 2$

$v(2) = 12(4) - 2 = 46\ \text{m/s}$

Example 2: a tossed keyring

A keyring is tossed vertically at 12 m/s. What is its velocity 0.50 s and 2.0 s later? Upward counts as positive; use g = 9.8 m/s².

Position: $y = 12t - 4.9t^2$ , so $v = \frac{dy}{dt} = 12 - 9.8t$ .

At 0.50 s: $v = 12 - 4.9 = 7.1\ \text{m/s}$ , moving up.

At 2.0 s: $v = 12 - 19.6 = -7.6\ \text{m/s}$ , moving down at 7.6 m/s.

Example 3: reading a tangent

On a position time graph, the tangent drawn at t = 4 s passes through (2 s, 6 m) and (6 s, 18 m). Find the instantaneous velocity at 4 s.

$v = \frac{18 - 6}{6 - 2} = \frac{12}{4} = 3.0\ \text{m/s}$

Using the tangent on a graph

Mark the moment you care about on the position time curve.
Draw a straight line that just touches the curve there, matching its direction.
Pick two points far apart on that line, so reading errors stay small.
Divide the rise in position by the run in time. Keep the sign.

A tangent sloping up means positive velocity, sloping down means negative, and a flat tangent at a peak or trough means the object is momentarily at rest.

Instantaneous versus average velocity

Comparing the two velocities
Feature	Instantaneous	Average
Covers	One instant	A time interval
Formula	$\frac{dx}{dt}$	$\frac{\Delta x}{\Delta t}$
Graph	Slope of tangent	Slope of chord
Equal when	Velocity is constant	Velocity is constant

For uniform motion the two agree at every moment. For anything else, the average velocity formula hides the detail that the instantaneous value shows. With constant acceleration you can skip calculus and use v = u + at from the final velocity formula page.

Where students go wrong

Using a chord instead of a tangent. A line through two points on the curve gives an average.
Differentiating the wrong function. Make sure you differentiate position, not velocity, to get velocity.
Dropping the sign. A negative result tells you the direction, so report it.
Confusing speed and velocity. The speedometer shows the size only.

The velocity formula page sets out the basic definition this builds on, and kinematics formulas gathers the full motion chapter.

Frequently asked questions

What is the formula of instantaneous velocity?

v = dx / dt, the derivative of position with respect to time. Without calculus, it is the slope of the tangent to the position time graph.

Is instantaneous velocity a vector?

Yes. It has a size, the instantaneous speed, and points in the direction the object is moving at that moment.

Can instantaneous velocity be zero while acceleration is not?

Yes. At the turning point of an upward toss the velocity passes through zero, yet gravity keeps pulling, so the acceleration there remains g, pointing down.

How is instantaneous speed related to instantaneous velocity?

Instantaneous speed is the size of the instantaneous velocity. For a moment in time, the two always have the same magnitude.

What is the unit of instantaneous velocity?

Metre per second (m/s), the same as any velocity.