Change in Velocity Formula: Δv = v − u, Δv = at, Examples

The change in velocity formula is $\Delta v = v - u$ : final velocity minus initial velocity. In US notation it reads $\Delta v = v_f - v_i$ . Because velocity is a vector, this is a vector subtraction, and the answer has both a size and a direction.

For a car speeding up along a straight road, the change in velocity is just the gain in speed. When an object reverses or turns, things get interesting: a ball that bounces back can have a change in velocity larger than either of its speeds. The change in velocity also equals $at$ for constant acceleration and $\frac{J}{m}$ when an impulse J acts on a mass m.

Quick facts

Formula
$\Delta v = v - u$
From acceleration
$\Delta v = at$
From impulse
$\Delta v = \frac{J}{m} = \frac{F\,\Delta t}{m}$
SI unit
m/s
Dimensional formula
[M⁰ L¹ T⁻¹]
Type
Vector, points the way the velocity was pushed

Change in velocity formula along a straight line

Choose a positive direction and give each velocity its sign. Then subtract final minus initial.

Change in velocity for common straight line cases
Situation	u	v	Δv
Speeding up forward	+12 m/s	+20 m/s	+8 m/s
Slowing down forward	+20 m/s	+5 m/s	−15 m/s
Bouncing back	−8 m/s	+6 m/s	+14 m/s
Stopping	+25 m/s	0	−25 m/s

A negative Δv does not always mean slowing down. It means the velocity has moved toward the negative direction. Read the sign together with the direction of motion.

Worked examples

Example 1: a car on a straight road

A car goes from 12 m/s to 20 m/s along a straight road. Find the change in velocity.

$\Delta v = 20 - 12 = 8\ \text{m/s forward}$

Example 2: a basketball bounce

A basketball hits the floor moving down at 8.0 m/s and leaves moving up at 6.0 m/s. Find the change in velocity. Take up as positive.

u = −8.0 m/s, v = +6.0 m/s.

$\Delta v = 6.0 - (-8.0) = 14\ \text{m/s upward}$

The floor had to reverse the ball and then push it back up, so the change equals the two speeds added together.

Example 3: a right angle turn

A hockey puck sliding east at 2.4 m/s is deflected and slides north at 3.2 m/s. Find the size and direction of the change in velocity.

Components: Δv_x = 0 − 2.4 = −2.4 m/s (west), Δv_y = 3.2 − 0 = 3.2 m/s (north).

$|\Delta v| = \sqrt{2.4^2 + 3.2^2} = \sqrt{16} = 4.0\ \text{m/s}$

Direction: $\tan^{-1}\left(\frac{3.2}{2.4}\right) \approx 53^\circ$ north of west.

Example 4: from impulse

A 0.15 kg ball at rest is struck with an impulse of 3.0 N s. What is its change in velocity?

$\Delta v = \frac{J}{m} = \frac{3.0}{0.15} = 20\ \text{m/s}$

Other ways to find the change in velocity

Three routes lead to the same Δv, and which one to use depends on the data:

From two velocities: $\Delta v = v - u$ , the definition.
From acceleration and time: $\Delta v = at$ when a is constant. A body accelerating at 1.5 m/s² for 8.0 s changes its velocity by 12 m/s, whatever it started with.
From force: Newton's second law in impulse form gives $m\Delta v = F\Delta t$ . See the impulse formula and change in momentum formula pages.

On a velocity time graph, the change in velocity between two times is the difference of the two vertical readings, and it also equals the area beneath an acceleration against time plot.

Why the vector view matters

Speed and velocity are different. A bus circling a ring road at a constant 10 m/s has zero change in speed, but its velocity keeps changing because its direction changes. After half a lap the velocity has reversed, and the change is 20 m/s. This is why a body moving in a circle is accelerating, the idea behind the centripetal acceleration formula. Dividing Δv by the time taken gives the average acceleration formula.

Mistakes with Δv

Subtracting speeds instead of velocities on a rebound. Signs turn 6 − 8 into 6 − (−8).
Reversing the order. Always final minus initial.
Adding perpendicular components directly. Use Pythagoras.
Calling a turn at constant speed zero change. The direction changed, so Δv is not zero.

The kinematics formulas hub and the velocity formula page give the background.

Frequently asked questions

What is the formula for change in velocity?

Δv = v − u, final velocity minus initial velocity, done as a vector subtraction. With constant acceleration, Δv = at.

Can change in velocity be larger than the speed?

Yes. When an object reverses, Δv equals the sum of the two speeds. A ball hitting a wall at 10 m/s and bouncing back at 10 m/s has Δv = 20 m/s.

Is change in velocity a vector?

Yes. It points the way the velocity was pushed, which is the same direction as the average acceleration and the net impulse.

How is change in velocity related to acceleration?

Average acceleration is Δv divided by the time taken. For constant acceleration, Δv = at.

What is the change in velocity for uniform circular motion over a full circle?

Zero, because the object is back to the same velocity. Over half a circle the change is twice the speed.