A motion graphs interpreter helps you connect the three main kinematics graphs:
position-time, velocity-time, and acceleration-time.
Each graph answers a different question. The position-time graph tells you where the object is,
the velocity-time graph tells you how fast and in what direction it moves,
and the acceleration-time graph tells you how the velocity is changing.
The most important idea is that these graphs are not isolated.
Their slopes and areas connect them directly.
1. Position-time graph
If the horizontal axis is time and the vertical axis is position,
then the value of the graph at a chosen time gives the position
\(x(t)\).
More importantly, the slope of the x-t graph gives the velocity:
\[
\begin{aligned}
v(t) &= \frac{dx}{dt}
\end{aligned}
\]
That is why the calculator draws a tangent line on the x-t graph when you click at a time.
A steep positive slope means a large positive velocity.
A flat slope means zero velocity.
A negative slope means the object is moving in the negative direction.
2. Velocity-time graph
The point value on the v-t graph is the instantaneous velocity.
But this graph also has an area interpretation.
The signed area under the velocity-time graph between two times gives the displacement:
\[
\begin{aligned}
\Delta x &= \int_{t_0}^{t_1} v(t)\,dt
\end{aligned}
\]
If the curve lies above the time axis, the area is positive.
If it lies below the time axis, the area is negative.
So the calculator shades the area under the v-t graph from the start time up to the selected click time,
and that shaded region is reported as the displacement from the start.
3. Acceleration-time graph
The point value on the a-t graph is the instantaneous acceleration.
The slope of the v-t graph is acceleration,
which means
\[
\begin{aligned}
a(t) &= \frac{dv}{dt}
\end{aligned}
\]
The acceleration-time graph also has an area interpretation.
The signed area under the a-t graph gives the change in velocity:
\[
\begin{aligned}
\Delta v &= \int_{t_0}^{t_1} a(t)\,dt
\end{aligned}
\]
This is why the calculator shades the area under the a-t graph up to the clicked time and reports it as the
velocity change from the start.
4. Constant acceleration model
When the acceleration is constant,
the equations are
\[
\begin{aligned}
a(t) &= a_0 \\
v(t) &= v_0 + a_0(t-t_0) \\
x(t) &= x_0 + v_0(t-t_0) + \tfrac{1}{2}a_0(t-t_0)^2
\end{aligned}
\]
In this case:
the a-t graph is a horizontal line,
the v-t graph is a straight line,
and the x-t graph is a parabola.
So if you see a straight line on the velocity graph, you should expect a curved position graph.
5. Linearly changing acceleration model
The calculator also supports a simple non-constant acceleration model:
\[
\begin{aligned}
a(t) &= a_0 + j(t-t_0)
\end{aligned}
\]
Here
\(j\)
is the rate at which acceleration changes with time.
Integrating once gives velocity,
and integrating again gives position:
\[
\begin{aligned}
v(t) &= v_0 + a_0(t-t_0) + \tfrac{1}{2}j(t-t_0)^2 \\
x(t) &= x_0 + v_0(t-t_0) + \tfrac{1}{2}a_0(t-t_0)^2 + \tfrac{1}{6}j(t-t_0)^3
\end{aligned}
\]
In that model, the a-t graph is a straight line,
the v-t graph becomes a parabola,
and the x-t graph becomes a cubic-like curve.
This gives a simple way to study non-constant acceleration without needing a fully general symbolic input.
6. Worked example
Use the sample values
\(x_0 = 0\),
\(v_0 = 5\,\mathrm{m\,s^{-1}}\),
\(a = -2\,\mathrm{m\,s^{-2}}\),
and
\(t = 0\) to \(5\,\mathrm{s}\).
Step 1. Velocity at \(t=5\)
\[
\begin{aligned}
v(5) &= v_0 + a(5-0) \\
&= 5 + (-2)(5) \\
&= -5\,\mathrm{m\,s^{-1}}
\end{aligned}
\]
Step 2. Position at \(t=5\)
\[
\begin{aligned}
x(5) &= x_0 + v_0(5-0) + \tfrac{1}{2}a(5-0)^2 \\
&= 0 + 5(5) + \tfrac{1}{2}(-2)(25) \\
&= 25 - 25 \\
&= 0\,\mathrm{m}
\end{aligned}
\]
Step 3. Graph interpretation
The final velocity is negative, so the slope of the x-t graph at \(t=5\) is negative.
The area under the a-t graph from \(0\) to \(5\) is
\(-10\,\mathrm{m\,s^{-1}}\),
which matches the change from
\(v_0 = 5\) to
\(v(5) = -5\).
The area under the v-t graph from \(0\) to \(5\) is zero,
which matches the net displacement \(x(5)-x_0 = 0\).
7. What the calculator shows
The animation displays a particle moving on a one-dimensional track.
The particle position matches the x-t graph,
the velocity arrow matches the x-t slope and v-t point value,
and the acceleration arrow matches the a-t point value.
When you click the graphs, the selected time updates across all panels.
The tangent line on the x-t graph, the shaded area under the v-t graph,
and the shaded area under the a-t graph all update together.
This makes the slope-and-area interpretation visually immediate.
Summary
| Graph |
Point value |
Slope |
Area |
| x-t |
\(x(t)\) |
\(v(t)\) |
No standard direct meaning used here |
| v-t |
\(v(t)\) |
\(a(t)\) |
\(\Delta x = \int v(t)\,dt\) |
| a-t |
\(a(t)\) |
Rate of change of acceleration |
\(\Delta v = \int a(t)\,dt\) |
