Linear motion with constant acceleration is one of the core models in introductory mechanics. The motion takes place along a single axis, so all quantities carry signs. A positive sign means motion or acceleration in the chosen positive direction, while a negative sign means the opposite direction.
Core kinematic equations
If the acceleration \(a\) is constant, the main one-dimensional kinematic relations are:
Velocity equation.
\[
\begin{aligned}
v &= v_0 + a t
\end{aligned}
\]
Position equation.
\[
\begin{aligned}
x &= x_0 + v_0 t + \tfrac{1}{2} a t^2
\end{aligned}
\]
Displacement equation.
\[
\begin{aligned}
\Delta x &= x - x_0
\end{aligned}
\]
Average-velocity form for constant acceleration.
\[
\begin{aligned}
\Delta x &= \frac{v_0 + v}{2}\,t
\end{aligned}
\]
Velocity-displacement relation.
\[
\begin{aligned}
v^2 &= v_0^2 + 2 a \Delta x
\end{aligned}
\]
These equations allow different known-variable sets to determine the remaining unknowns, which is why the calculator provides several solve modes instead of only one fixed input pattern.
How the solver chooses equations
The idea is simple: pick the equation that contains the unknown quantity and the available knowns, then solve in a consistent order. For example, if \(v_0\), \(a\), and \(t\) are known, the velocity equation gives \(v\) immediately, and the position equation then gives \(x\). If the final position \(x\) is known instead of time, the calculator may need to solve a quadratic equation in \(t\).
Example of a quadratic time equation.
\[
\begin{aligned}
\Delta x &= v_0 t + \tfrac{1}{2} a t^2 \\
0 &= \tfrac{1}{2} a t^2 + v_0 t - \Delta x
\end{aligned}
\]
A quadratic can produce two real times, one real time, or no real time. If two nonnegative times are possible, the calculator reports that situation and uses the smallest nonnegative time as the default physical branch.
Worked prompt example
Use the prompt values
\(x_0 = 0\),
\(v_0 = 3\,\mathrm{m\,s^{-1}}\),
\(a = -1.5\,\mathrm{m\,s^{-2}}\),
and
\(t = 4\,\mathrm{s}\).
Step 1. Compute the final velocity.
\[
\begin{aligned}
v &= v_0 + a t \\
&= 3 + (-1.5)(4) \\
&= 3 - 6 \\
&= -3\,\mathrm{m\,s^{-1}}
\end{aligned}
\]
Step 2. Compute the final position.
\[
\begin{aligned}
x &= x_0 + v_0 t + \tfrac{1}{2} a t^2 \\
&= 0 + 3(4) + \tfrac{1}{2}(-1.5)(4^2) \\
&= 12 - 12 \\
&= 0\,\mathrm{m}
\end{aligned}
\]
Step 3. Compute the displacement.
\[
\begin{aligned}
\Delta x &= x - x_0 \\
&= 0 - 0 \\
&= 0\,\mathrm{m}
\end{aligned}
\]
This example is mathematically consistent: the object starts at the origin, moves forward, slows down, reverses direction, and returns to the starting point at \(t=4\ \mathrm{s}\). The velocity is negative at that instant because the object is moving back in the negative direction.
Why signs matter
In one-dimensional motion, the sign convention is essential. A negative acceleration does not automatically mean the object is slowing down. It only means the acceleration points in the negative direction. Whether the speed increases or decreases depends on the combination of the signs of \(v\) and \(a\).
| Case |
Direction of motion |
What happens to speed |
| \(v>0,\ a>0\) |
Positive |
Speed increases |
| \(v>0,\ a<0\) |
Positive |
Speed decreases until possible reversal |
| \(v<0,\ a<0\) |
Negative |
Speed increases in the negative direction |
| \(v<0,\ a>0\) |
Negative |
Speed decreases until possible reversal |
What the graph and animation show
The position-time graph is generally a parabola under constant acceleration, while the velocity-time graph is a straight line. The animation shows the same motion along a track, together with velocity and acceleration arrows. This gives a useful check: the signs and trends in the graphs should match the direction of the moving block and its arrows.
Summary
| Quantity |
Formula |
Meaning |
| Final velocity |
\(v = v_0 + a t\) |
Velocity after time \(t\) |
| Final position |
\(x = x_0 + v_0 t + \tfrac{1}{2} a t^2\) |
Position after time \(t\) |
| Displacement |
\(\Delta x = x - x_0\) |
Change in position |
| Average-velocity form |
\(\Delta x = \dfrac{v_0+v}{2} t\) |
Useful when both endpoint velocities are known |
| Velocity-displacement relation |
\(v^2 = v_0^2 + 2 a \Delta x\) |
Connects speeds without explicit time |
