Constant Acceleration in 1D

Constant acceleration means the acceleration stays the same throughout the motion interval. In one dimension, that makes the velocity change at a steady rate and turns the velocity-time graph into a straight line. The sign matters: positive values follow your chosen positive direction, and negative values point the opposite way.

The Four Kinematic Equations

These four equations describe the same constant-acceleration motion from different angles:

1. $v = v_0 + at$

Use this when you know initial velocity, acceleration, and time, or when you want the rate-of-change relationship directly.

1. $s = v_0t + \tfrac{1}{2}at^2$

Use this when displacement depends on how long the acceleration acts.

1. $v^2 = v_0^2 + 2as$

Use this when time is missing and you want to connect velocity change directly to displacement.

1. $s = \tfrac{1}{2}(v_0 + v)t$

Use this when you know the starting and ending velocities and want the average-velocity view of displacement.

Because the acceleration is constant, the equations are consistent with one another. A good strategy is to pick the equation that contains your unknown and avoids introducing an extra missing variable.

Why Displacement Is the Area Under the Velocity-Time Graph

On a velocity-time graph, the horizontal axis is time and the vertical axis is velocity. The area under the curve represents velocity multiplied by time, which has units of displacement.

• If $v_0 = 0$, the shaded region is a triangle, so displacement is:

• If $v_0 \neq 0$, the region becomes a trapezoid, so displacement is:

That is why the graph on this page is so useful: it turns the algebra into a picture. The slope of the line is the acceleration, and the shaded area is the displacement.

Real-World Examples

• Free fall: Near Earth’s surface, a falling object has an almost constant downward acceleration of about $9.8\ \text{m/s}^2$ if air resistance is small.
• Braking: A car that slows at a nearly steady rate has approximately constant negative acceleration, which makes the velocity-time graph slope downward.
• Rocket launch phase: Over a short interval, a rocket can often be modeled with roughly constant net acceleration, especially in simplified classroom problems.

How To Use This Calculator

1. Choose the quantity you want to solve for: final velocity $v$, displacement $s$, time $t$, or acceleration $a$.
2. Enter the three known quantities shown in the input card.
3. Watch the motion diagram update to show which variable is known, unknown, or derived.
4. Read the result summary for the full set of kinematic quantities.
5. Use the step-by-step derivation and velocity-time graph to interpret the result physically, not just numerically.

If a result looks surprising, check the signs of the values you entered. A negative acceleration often represents braking, and a negative velocity means the object is moving in the opposite direction from the positive axis you chose.