What are tangential and normal components of acceleration in curvilinear motion?

In one line

On a curved path, acceleration splits into a tangential part that changes speed and a normal part, towards the centre of curvature, that changes direction.

Explained simply

When a bike speeds up while going round a bend, you feel pushed back (speeding up) and pushed sideways (turning). The backward feeling comes from tangential acceleration, the sideways one from normal (centripetal) acceleration.

Technically, with the unit tangent \( \hat{e}_t \) and unit normal \( \hat{e}_n \) towards the centre of curvature: \( \vec{a} = \dfrac{dv}{dt}\hat{e}_t + \dfrac{v^2}{\rho}\hat{e}_n \).

How it works

1. Find speed v and the rate of change of speed \( dv/dt \): this is \( a_t \).
2. Find the radius of curvature \( \rho \) of the path at that point (given, or from the path).
3. \( a_n = v^2/\rho \), always directed towards the centre of curvature.
4. Total \( a = \sqrt{a_t^2 + a_n^2} \).
5. Special cases: uniform circular motion: \( a_t = 0 \); straight line: \( \rho \to \infty \), \( a_n = 0 \).
6. For circular motion with angular speed \( \omega \) and angular acceleration \( \alpha \): \( a_t = r\alpha \), \( a_n = r\omega^2 \).

Key formula

\( a_t = \dfrac{dv}{dt} \), \( a_n = \dfrac{v^2}{\rho} \), \( a = \sqrt{a_t^2 + a_n^2} \). v in m/s, \( \rho \) in m, a in m/s\(^2\).

Quick example

A car on a curve of radius 100 m moves at 20 m/s and is speeding up at 2 m/s\(^2\). \( a_n = 400/100 = 4 \) m/s\(^2\), \( a_t = 2 \) m/s\(^2\), total \( a = \sqrt{20} = 4.47 \) m/s\(^2\), at \( \tan^{-1}(4/2) = 63.4^\circ \) to the tangent.

Common mistakes

• Writing \( a_t = v^2/\rho \) and \( a_n = dv/dt \) (swapped).
• Thinking a particle at constant speed has zero acceleration on a curve.
• Taking \( a_t = dv/dt \) by differentiating a component instead of the speed.

How MU asks it

Theory 3 marks: 'Explain tangential and normal components of acceleration'. Numerical 5 to 6 marks: car or train on a curved track with changing speed, find total acceleration. Draw the path, show \( a_t \) along the tangent and \( a_n \) towards the centre, then combine.