A car starts at 4 m/s and decelerates at -2 m/s². When does it come to a stop?

The correct answer is: 2 s. You can work it out this way: set v = 0 in v = v₀ + at. Solve for t.

What does a negative area under a v-t graph represent physically?

The correct answer is: The object moved in the negative direction. Area = displacement. Negative area means the object moved in the negative direction.

Sketch what an a-t graph looks like for constant acceleration motion. What shape is it?

The correct answer is: A horizontal line. If acceleration is constant, what does its graph vs. time look like?

Pro 🔒~20 min

Kinematics Graphs

Interpret position-time and velocity-time graphs for uniform acceleration

How it works

Kinematics describes motion without asking why it happens. For constant acceleration, position follows a parabola on an x-t graph while velocity traces a straight line on a v-t graph. The slope of the x-t curve at any instant equals the instantaneous velocity; the slope of the v-t line equals acceleration. Crucially, the signed area under the v-t graph between two times equals the displacement — not the distance. Negative velocity means motion in the negative direction; negative area means the object moved backward. Mastering these graph relationships is the single most tested skill on AP Physics 1.

Can an object have zero velocity but non-zero acceleration?

Think about a ball thrown straight up at the very top of its arc.

What you'll learn

Position-Time Graphs. A position-time graph shows where an object is at each moment. For constant acceleration, the curve is a parabola. The slope at any point equals the instantaneous velocity — steeper slope means faster motion.
Velocity-Time Graphs. A velocity-time graph for constant acceleration is always a straight line. The slope of this line equals the acceleration, and the area under the line equals displacement — not distance traveled.
Acceleration-Time Graphs. For uniform acceleration, the a-t graph is a horizontal line. The area under the a-t curve over a time interval equals the change in velocity during that interval. Zero area means the speed did not change.
Connecting the Three Graphs. The three motion graphs are mathematically linked: the slope of x-t gives v-t, and the slope of v-t gives a-t. Working backwards, the area under a-t gives change in v, and the area under v-t gives displacement. Mastering these connections is the single most tested skill on AP Physics 1.

Step-by-step

Adjust initial velocity and acceleration sliders.
The animation shows a ball moving along a track while both graphs update in real time.
Pause at any moment to read the exact x and v values.
Try v₀ = 0, a = 2 to get pure parabolic x-t and linear v-t.
Then try v₀ = 5, a = -2 — note the x-t maximum occurs exactly where v-t crosses zero.
Toggle Graph Mode (Pro) to focus on one graph at a time.

Key formulas

$x = x_0 + v_0 t + \frac{1}{2}at^2$ Position as a function of time (uniform acceleration)
$v = v_0 + at$ Velocity as a function of time
$v^2 = v_0^2 + 2a\Delta x$ Velocity-displacement kinematic equation
$\Delta x = \text{area under } v\text{-}t \text{ graph}$ Displacement equals the area under the velocity-time graph

Frequently asked questions

A car starts at 4 m/s and decelerates at -2 m/s². When does it come to a stop?: The correct answer is: 2 s. You can work it out this way: set v = 0 in v = v₀ + at. Solve for t.
What does a negative area under a v-t graph represent physically?: The correct answer is: The object moved in the negative direction. Area = displacement. Negative area means the object moved in the negative direction.
Sketch what an a-t graph looks like for constant acceleration motion. What shape is it?: The correct answer is: A horizontal line. If acceleration is constant, what does its graph vs. time look like?