Constant Acceleration Formula: The Four Kinematic Equations

The constant acceleration formula used in AP Physics is $x = x_0 + v_0 t + \frac{1}{2}at^2$ . It gives the position x of an object at time t when it starts at $x_0$ with velocity $v_0$ and its acceleration a never changes. Three partner equations complete the set, and together they are called the kinematic equations.

Indian books write the same ideas with u for $v_0$ and s for $x - x_0$ , and call them the equations of motion. The physics is identical. What matters is the condition in the name: the acceleration may not grow, shrink or turn at any point in the interval, as in free fall or steady braking.

Quick facts

Position
$x = x_0 + v_0 t + \frac{1}{2}at^2$
Velocity
$v = v_0 + at$
No time
$v^2 = v_0^2 + 2a(x - x_0)$
No acceleration
$x - x_0 = \frac{(v_0 + v)}{2}t$
Unit of a
m/s², dimension [M⁰ L¹ T⁻²]
Graph sign
Straight line on a velocity time graph

Constant acceleration formula physics: the four kinematic equations

The kinematic equations in two notations
US form	Indian form	Missing quantity
$v = v_0 + at$	$v = u + at$	displacement
$x = x_0 + v_0 t + \frac{1}{2}at^2$	$s = ut + \frac{1}{2}at^2$	final velocity
$v^2 = v_0^2 + 2a(x - x_0)$	$v^2 = u^2 + 2as$	time
$x - x_0 = \frac{(v_0 + v)}{2}t$	$s = \frac{(u + v)}{2}t$	acceleration

Pick the equation that leaves out the quantity you neither know nor need. The x₀ term in the position form is a convenience: it lets you place the object anywhere on an axis, which helps when two objects start at different points.

How to tell the acceleration is constant

The net force on the object is constant, for example gravity alone near Earth's surface.
The velocity time graph is a straight line. Its slope is a.
The position time graph is a parabola, curving up for positive a and down for negative a.
The acceleration time graph is a flat horizontal line.

If none of these hold, as for a car with changing throttle or a falling object with strong air resistance, the kinematic equations give wrong answers.

Worked examples

Example 1: a cart on a track

A cart starts at x₀ = 2.0 m with v₀ = 3.0 m/s and a constant acceleration of −0.50 m/s². Where is it at t = 4.0 s?

$x = 2.0 + 3.0(4.0) + \tfrac{1}{2}(-0.50)(4.0)^2 = 2.0 + 12 - 4.0 = 10\ \text{m}$

Example 2: a road test quoted in km/h

A car goes from rest to 100 km/h in 8.0 s with steady acceleration. Find the acceleration and the distance covered.

100 ÷ 3.6 ≈ 27.8 m/s.

$a = \frac{27.8 - 0}{8.0} \approx 3.47\ \text{m/s}^2$

$x - x_0 = \frac{(0 + 27.8)}{2} \times 8.0 \approx 111\ \text{m}$

Example 3: a ball thrown up from a height

A ball leaves a hand 1.5 m above the ground moving upward at 8.0 m/s. When does it land? Measure x upward from the ground.

$0 = 1.5 + 8.0t - 4.9t^2$ , so $4.9t^2 - 8.0t - 1.5 = 0$ .

$t = \frac{8.0 + \sqrt{64 + 29.4}}{9.8} = \frac{8.0 + 9.66}{9.8} \approx 1.80\ \text{s}$

The other root is negative and has no physical meaning here.

A dimension check on the position formula

Every term must be a length. $x_0$ is in m. $v_0 t$ is (m/s)(s) = m. $\frac{1}{2}at^2$ is (m/s²)(s²) = m. The ½ has no dimensions, so a dimension check cannot tell you whether it belongs there; only the derivation can. For the acceleration itself, the dimensions are [M⁰ L¹ T⁻²], as set out on the acceleration formula page.

Traps

Sign of g. If x increases upward, the acceleration is −9.8 m/s² throughout the flight, rising and falling alike.
Splitting the motion when you do not need to. One equation covers the up and down path as long as a stays constant.
Keeping the wrong root of a quadratic in t.
Mixing notations. Do not write u in one line and v₀ in the next.

The u and s versions live on the formula of motion page, v = v₀ + at gets its own treatment as the final velocity formula, and projectile uses are listed under kinematics formulas.

Frequently asked questions

What is the constant acceleration formula?

The main one is x = x₀ + v₀t + ½at². Its partners are v = v₀ + at, v² = v₀² + 2a(x − x₀) and x − x₀ = (v₀ + v)t / 2.

Is free fall an example of constant acceleration?

Yes, provided drag is negligible. The acceleration is g = 9.8 m/s² downward the whole time.

Can the velocity be zero while acceleration is constant?

Yes. A ball thrown upward has zero velocity at the top, but its acceleration is still g downward.

What does a constant acceleration look like on a graph?

Velocity against time is a tilted straight line, position against time is a parabola, and acceleration against time is level.

Are the kinematic equations the same as the equations of motion?

Yes. US courses call them kinematic equations and write v₀ and Δx. Indian courses call them equations of motion and write u and s.