Final Velocity Formula: v = u + at and v² = u² + 2as

The final velocity formula is $v = u + at$ : start with the initial velocity u and add the velocity gained from an acceleration a acting for a time t. It is the first equation of motion, and it holds whenever the acceleration stays constant.

US books write it $v_f = v_i + at$ or $v = v_0 + at$ . If a question gives distance instead of time, use the third equation, $v^2 = u^2 + 2as$ , which finds the final velocity without ever needing t. Choosing between the two is usually the first decision in a straight line motion problem.

Quick facts

Formula
$v = u + at$
Without time
$v = \sqrt{u^2 + 2as}$
SI unit
m/s
Dimensional formula
[M⁰ L¹ T⁻¹]
Condition
Acceleration must be constant
Free fall from rest
$v = gt$

Final velocity formula physics: what each letter means

Symbols in v = u + at
Symbol	Quantity	SI unit	US symbol
$v$	Final velocity	m/s	$v_f$
$u$	Initial velocity	m/s	$v_i$ or $v_0$
$a$	Constant acceleration	m/s²	$a$
$t$	Time the acceleration acts	s	$t$

The product at is the change in velocity. A car accelerating at 2 m/s² for 5 s gains 10 m/s, so whatever it started with, it ends 10 m/s faster. The formula comes straight from the definition of acceleration, $a = \frac{v - u}{t}$ , multiplied out.

Formula for final velocity: worked examples

Example 1: a bicycle speeding up

A cyclist moving at 3.0 m/s accelerates at 0.80 m/s² for 10 s. Find the final velocity.

$v = 3.0 + 0.80 \times 10 = 3.0 + 8.0 = 11\ \text{m/s}$

Example 2: free fall

A coconut drops from rest. What is its velocity after 3.0 s? Use g = 9.8 m/s² and ignore air resistance.

$v = 0 + 9.8 \times 3.0 = 29.4\ \text{m/s downward}$

Example 3: no time given, with km/h

A car moving at 36 km/h accelerates uniformly at 1.5 m/s² over 75 m. Find its final velocity.

u = 36 ÷ 3.6 = 10 m/s. No time is given, so use the third equation.

$v^2 = 10^2 + 2(1.5)(75) = 100 + 225 = 325$

$v = \sqrt{325} \approx 18.0\ \text{m/s} \approx 64.9\ \text{km/h}$

Example 4: braking

A scooter at 12 m/s brakes with a retardation of 3.0 m/s². What is its velocity after 2.5 s?

Retardation means a = −3.0 m/s². $v = 12 + (-3.0)(2.5) = 12 - 7.5 = 4.5\ \text{m/s}$ .

Picking the right equation

Which final velocity equation fits the question
Known	Missing	Use
u, a, t	s	$v = u + at$
u, a, s	t	$v^2 = u^2 + 2as$
u, s, t	a	$v = \frac{2s}{t} - u$

The last row comes from $s = \frac{(u + v)}{2}t$ . List what you know and what you need before you pick. The formula of motion page shows all three equations of motion together, and the constant acceleration formula page explains why they need a steady a.

The velocity time graph

Plot v = u + at and you get a straight line. The intercept on the velocity axis is u, and the slope is a. Read off the line at any time t and you have the final velocity at that moment. The area under the line up to that time gives the displacement, which is how the second equation of motion is derived in Class 9 books. If the line crosses the time axis, the object has stopped and is about to move the other way, which is exactly what happens to a ball thrown upward.

Mistakes with final velocity

Sign of a. For a slowing object, a is negative when the motion is positive.
Square root at the end. v² = u² + 2as gives v², not v.
Negative under the root. If u² + 2as is negative, the object stopped before covering s.
Variable acceleration. None of these formulas apply. Use calculus or a graph.
Mixed units. A speed in km/h with an acceleration in m/s² gives a wrong v. Convert to m/s first.

The opposite problem, working back to the starting velocity, is on the initial velocity formula page. The kinematics formulas hub links the rest.

Frequently asked questions

What is the formula for final velocity?

Final velocity = initial velocity + acceleration × time, or v = u + at. When the question gives a distance rather than a time, switch to v² = u² + 2as.

How do you find final velocity without time?

Use v² = u² + 2as, then take the square root. It needs the initial velocity, the acceleration and the displacement.

What is the final velocity of a freely falling body?

Starting from rest, v = gt after time t, or v = √(2gh) after falling a height h, ignoring air resistance.

Can final velocity be less than initial velocity?

Yes. If the acceleration is opposite to the motion, as in braking, the final velocity is smaller, and it can even become negative.

Is v = u + at valid for all motion?

No. If the push on an object grows, fades or swings round, the formula gives wrong answers. Use it only for steady, unchanging acceleration.