The same physical motion can be described in several equivalent ways. Each representation highlights something different, and a strong physics student can move between them without losing track of the underlying motion.
A verbal story: 'the car speeds up from rest, then cruises.'
Position or velocity listed at specific times.
Dots showing position at equal time intervals — spacing shows speed.
x-t, v-t, and a-t curves that show the whole motion at a glance.
For motion with constant acceleration, the three graphs always have the same shapes. Reading left to right, each graph is the slope of the one before it.
Curves (parabola). Its slope at any instant is the velocity.
A straight, sloped line. Its slope is the acceleration; its area is displacement.
A flat horizontal line — acceleration is constant.
The slope of a graph is "rise over run" — the change in the vertical quantity divided by the change in time. That makes slope a rate.
So the slope of a position-time graph is velocity, and the slope of a velocity-time graph is acceleration. Steeper slope means a faster rate of change; a flat line means the quantity isn't changing.
Slope works backward too. Going the other direction — from a velocity-time graph back to position — you use the area under the curve. The area under a velocity-time graph is the displacement; the area under an acceleration-time graph is the change in velocity.
Use the explorer below. Change the acceleration and watch the slope of the line change. Change either slider and watch the shaded area — the displacement — update.
Set the initial velocity and acceleration. The slope of the line is the acceleration; the shaded area under it is the displacement.
When acceleration is constant, three equations connect position, velocity, acceleration, and time. Each one is missing a different variable — pick the equation that doesn't contain the quantity you don't know and aren't solving for.
| Equation | Missing variable | Use when… |
|---|---|---|
v = v₀ + at | Δx (position) | You know acceleration and time, want final velocity. |
x = x₀ + v₀t + ½at² | v (final velocity) | You want position after a known time. |
v² = v₀² + 2aΔx | t (time) | Time isn't given and isn't asked for. |
A car starts from rest and accelerates at 4 m/s² for 6 s. How far does it travel?
Free fall is just constant-acceleration motion where the acceleration is gravity. Near Earth's surface, ignoring air resistance, every freely falling object accelerates downward at the same rate.
That means the kinematic equations apply directly — just substitute a = g (with the correct sign for your chosen direction). A ball thrown up and a rock dropped down obey the very same equations.
A ball is dropped from rest off a 45 m cliff. Using g ≈ 10 m/s², how long does it take to reach the ground, and how fast is it moving on impact?