Initial Velocity Formula: u = v − at and Other Forms

The initial velocity formula is $u = v - at$ . If you know how fast something is moving at the end, its constant acceleration and how long it accelerated, subtract the velocity it gained, at, and you are left with the velocity it started with.

Rearranging v = u + at, the first of the three equations, gives this directly. When the time is missing, the third equation gives $u = \sqrt{v^2 - 2as}$ , and when the final velocity is missing, the second gives $u = \frac{s - \frac{1}{2}at^2}{t}$ . Indian books call the starting velocity u. US books write $v_0$ or $v_i$ .

Quick facts

Formula
$u = v - at$
No time given
$u = \sqrt{v^2 - 2as}$
No final velocity given
$u = \frac{s}{t} - \frac{1}{2}at$
SI unit
m/s
US symbol
$v_0$ or $v_i$
Condition
Constant acceleration

Formula for initial velocity in three forms

Choosing an initial velocity formula
You know	Formula	From
v, a, t	$u = v - at$	First equation of motion
v, a, s	$u = \sqrt{v^2 - 2as}$	Third equation of motion
s, a, t	$u = \frac{s}{t} - \frac{1}{2}at$	Second equation of motion
s, v, t	$u = \frac{2s}{t} - v$	Average velocity form

All four assume the acceleration stays constant. Pick the row whose known quantities match the question, then substitute with signs. The full set of parent equations is on the formula of motion page.

Initial velocity formula examples

Example 1: a car after accelerating

A car reaches 25 m/s after accelerating at 2.5 m/s² for 4.0 s. What was its starting velocity?

$u = 25 - 2.5 \times 4.0 = 25 - 10 = 15\ \text{m/s}$

Example 2: braking distance

Skid marks show a car stopped in 50 m while decelerating steadily at 4.0 m/s². What speed was it doing when the brakes went on?

Here the final velocity is zero, $a = -4.0$ m/s² and $s = 50$ m. $u^2 = v^2 - 2as = 0 - 2(-4.0)(50) = 400$ .

$u = 20\ \text{m/s} = 72\ \text{km/h}$

Example 3: a throw that reaches a given height

A stone thrown vertically just reaches 20 m above the hand. Ignoring drag, find the launch speed (g = 9.8 m/s²).

At the top v = 0. With up positive, a = −9.8 m/s². $u^2 = 0 + 2(9.8)(20) = 392$ .

$u = \sqrt{392} \approx 19.8\ \text{m/s}$

Initial velocity in projectile problems

In projectile motion the launch velocity u is split into $u\cos\theta$ sideways and $u\sin\theta$ upward. If the range R and angle are known, rearranging the range formula gives $u = \sqrt{\frac{gR}{\sin 2\theta}}$ . Say a ball lands 40.8 m away after a 45° launch on level ground Since sin 90° = 1, it left at about $\sqrt{9.8 \times 40.8} \approx 20$ m/s. Knowing u, you can then find how long the ball was in the air and how high it rose, because both depend only on u, θ and g. The range of projectile formula and time of flight formula pages show the full method.

Signs and units

Choose a positive direction before you start, and write it next to the diagram so you do not change your mind halfway. If the object was moving the other way at the beginning, u comes out negative, and that is a correct and useful answer. Keep velocities in m/s and acceleration in m/s². A speed given in km/h must be divided by 3.6 first. The dimensions of initial velocity are the same as any velocity, [M⁰ L¹ T⁻¹], and every term in u = v − at has those dimensions.

Slips that cost marks

Adding at instead of subtracting. u = v − at; check by putting your u back into v = u + at.
Wrong sign for retardation. A slowing object has a opposite to its motion.
Forgetting the square root after using v² = u² + 2as.
Ignoring the second root. u = ±√(...) and the sign comes from the direction of motion.

Forward problems, finding v from u, are on the final velocity formula page, and every other straight line result is gathered under kinematics formulas.

Frequently asked questions

How do you write the formula for initial velocity?

u = v − at, from the first equation of motion. Without time, use u = √(v² − 2as).

How do you find initial velocity without time?

Use the third equation of motion rearranged: u² = v² − 2as. Take the square root and choose the sign that matches the direction of motion.

What is the initial velocity of a body dropped from rest?

Zero. Dropped means released without a push, so u = 0.

What symbol is used for initial velocity?

Indian books use u. US books use v₀ or vᵢ. They mean the same thing.

Can initial velocity be negative?

Yes. A negative u means the object was moving in the direction you chose as negative at the start.