In non-uniform circular motion, an object moves on a circular path of fixed radius
\(r\), but its speed is not constant. That means the acceleration must do two different jobs at the same time.
One part changes the magnitude of the velocity, while the other changes only its direction.
These two parts are called the tangential acceleration and the radial or
centripetal acceleration.
Tangential acceleration
The tangential component measures how quickly the speed changes along the path. If the speed changes from
\(v_i\) to \(v_f\) during a time interval \(\Delta t\), then the average tangential acceleration is
\[
\begin{aligned}
a_t &= \frac{\Delta v}{\Delta t} = \frac{v_f - v_i}{\Delta t}
\end{aligned}
\]
When \(a_t > 0\), the object speeds up. When \(a_t < 0\), the object slows down. The tangential-acceleration
vector lies along the tangent to the circle. If the object is speeding up, it points in the same direction as
the motion. If the object is slowing down, it points opposite to the motion.
Radial or centripetal acceleration
The radial component exists because the velocity direction is continuously turning toward the center of the
circle. It does not change how fast the object moves along the path; it changes only the direction of the
velocity vector. Its magnitude is
\[
\begin{aligned}
a_r &= \frac{v^2}{r}
\end{aligned}
\]
This means the radial acceleration depends on the instantaneous speed. If the speed changes with time,
then \(a_r\) also changes with time. At the initial instant, \(a_{r,i} = v_i^2/r\), and at the final instant,
\(a_{r,f} = v_f^2/r\).
Total acceleration
The tangential and radial components are perpendicular to each other. Because of that, the magnitude of the
total acceleration is found with the Pythagorean relation
\[
\begin{aligned}
|\vec a| &= \sqrt{a_t^2 + a_r^2}
\end{aligned}
\]
This is why the total acceleration is generally a little larger than the biggest component by itself. If the
radial part is dominant, the total-acceleration vector points mostly inward. If the tangential part is larger,
the total vector leans more along the tangent.
Time-varying speed model used by this calculator
This calculator assumes the speed changes uniformly during the chosen interval. That means the speed function is
\[
\begin{aligned}
v(t) &= v_i + a_t t
\end{aligned}
\]
Therefore, the radial acceleration also changes with time as
\[
\begin{aligned}
a_r(t) &= \frac{v(t)^2}{r}
\end{aligned}
\]
and the total acceleration becomes
\[
\begin{aligned}
|\vec a(t)| &= \sqrt{a_t^2 + a_r(t)^2}
\end{aligned}
\]
Arc length and angular motion
Since the motion is still on a circle, we can connect the linear quantities to angular ones. If the speed changes
uniformly, then the average speed over the interval is
\((v_i + v_f)/2\), so the arc length traveled is
\[
\begin{aligned}
s &= \left(\frac{v_i + v_f}{2}\right)\Delta t
\end{aligned}
\]
Dividing by the radius gives the angular displacement
\[
\begin{aligned}
\varphi &= \frac{s}{r}
\end{aligned}
\]
The corresponding angular speeds are
\(\omega_i = v_i/r\) and \(\omega_f = v_f/r\), and the angular acceleration is
\(\alpha = a_t/r\).
Worked example
Use the sample values
\(r = 10\,\mathrm{m}\),
\(v_i = 0\,\mathrm{m\,s^{-1}}\),
\(v_f = 20\,\mathrm{m\,s^{-1}}\),
and
\(\Delta t = 2\,\mathrm{s}\).
Step 1. Tangential acceleration
\[
\begin{aligned}
a_t &= \frac{20 - 0}{2} = 10\,\mathrm{m\,s^{-2}}
\end{aligned}
\]
Step 2. Final radial acceleration
\[
\begin{aligned}
a_{r,f} &= \frac{20^2}{10} = 40\,\mathrm{m\,s^{-2}}
\end{aligned}
\]
Step 3. Final total acceleration
\[
\begin{aligned}
|\vec a|_f &= \sqrt{10^2 + 40^2} \\
&= \sqrt{1700} \\
&\approx 41.2\,\mathrm{m\,s^{-2}}
\end{aligned}
\]
Notice that the final unit is m/s², not m/s. This is an acceleration, not a speed.
What the animation and graphs show
The animation keeps the velocity, tangential acceleration, radial acceleration, and total acceleration visible at
the same time. That makes the decomposition very clear: the velocity is tangent to the path, the radial
acceleration points inward, and the total acceleration is the vector sum of the radial and tangential parts.
The graphs then show the same physics in time form. The speed graph is linear, the tangential acceleration graph
is constant, the radial acceleration graph curves upward or downward depending on the speed trend, and the total
acceleration graph follows from the vector combination of the two components.
Summary
| Quantity |
Formula |
Meaning |
| Tangential acceleration |
\(a_t = (v_f - v_i)/\Delta t\) |
Changes the speed along the path |
| Radial acceleration |
\(a_r = v^2/r\) |
Changes the direction of motion toward the center |
| Total acceleration |
\(|\vec a| = \sqrt{a_t^2 + a_r^2}\) |
Vector combination of the perpendicular components |
| Arc length |
\(s = ((v_i+v_f)/2)\Delta t\) |
Distance traveled along the circular path |
| Angular displacement |
\(\varphi = s/r\) |
Total angle swept during the interval |
| Angular acceleration |
\(\alpha = a_t/r\) |
Angular version of the tangential speed change |
