NDA Physics · Kinematics and Motion

Circular Motion

In uniform circular motion the speed is constant but the velocity changes continuously because its direction keeps turning; this change is a centripetal acceleration of magnitude v²/r directed toward the centre.

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Why this matters

Three PYQs, but they punch above their weight because they expose the chapter's deepest idea: constant speed is NOT constant velocity. The bank tests that a body going round a circle at steady speed is still accelerating (toward the centre), that a gentler curve (larger radius) means a smaller acceleration, and — in a HARD item — the average acceleration over half a circle. Hold v²/r and the velocity-is-a-vector idea and these are quick marks.

Concept 1 of 3

Uniform circular motion — constant speed, changing velocity

Intuition

Drive a car round a roundabout at a steady 30 km/h: the speedometer never moves, but the car is constantly changing direction. Since velocity includes direction, the velocity is changing every instant — and a changing velocity means there is an acceleration, even though the speed is fixed.

Definition

In uniform circular motion the speed is constant but the velocity changes continuously because its direction is always changing (the velocity is tangent to the circle). A changing velocity means a non-zero acceleration; that acceleration points toward the centre and is called centripetal acceleration.

Worked example

A car moves uniformly along a circular track at constant speed. What, if anything, is changing about its motion?

Practice this conceptself-check · 4 quick reps

From the bank · past-year question

Example 1Kinematics and MotionEASY

A uniform motion of a car along a circular path experiences

[Q118 · Sep · 2021]

Constant speed is not constant velocity

The single most common error in circular motion: treating 'uniform speed' as 'no acceleration'. The direction of velocity changes every instant, so the velocity is changing and there IS an acceleration — directed toward the centre.

Concept 2 of 3

Centripetal acceleration — v²/r toward the centre

Intuition

The inward acceleration that keeps a body on its circular path has magnitude v²/r. For the same speed, a sharper curve (smaller radius) needs a bigger inward acceleration; a gentle curve (large radius) needs only a small one. That is why fast bends on a road are built with large radii.

Definition

The centripetal acceleration of a body moving at speed $v$ on a circle of radius $r$ has magnitude $a_c = \dfrac{v^2}{r}$ , directed toward the centre. At fixed speed it is inversely proportional to the radius: a gentle curve (large $r$ ) gives a smaller acceleration than a sharp curve (small $r$ ).

Centripetal acceleration

a_c = \dfrac{v^2}{r}

a_ccentripetal acceleration (toward centre)
vspeed
rradius of the circular path

Worked example

A car takes a bend of radius 50 m at 10 m/s. Find its centripetal acceleration.

Practice this conceptself-check · 4 quick reps

From the bank · past-year question

Example 2Kinematics and MotionMODERATE

An object moves in a circular path with a constant speed. Which one of the following statements is correct?

[Q69 · Sep · 2017]

Centripetal acceleration does not change the speed

The centripetal acceleration is perpendicular to the velocity (it points inward, velocity is tangent), so it changes only the DIRECTION of motion, never the speed. An option claiming 'centripetal acceleration causes the object to slow down' is false.

Concept 3 of 3

Average acceleration over part of a circle

Intuition

Centripetal acceleration v²/r is the INSTANTANEOUS value. The AVERAGE acceleration over a stretch of the path is different — it uses the actual change in the velocity vector divided by the time. Over half a circle the velocity simply reverses, so the change in velocity is 2v.

Definition

Average acceleration over an interval is $\bar{a} = \dfrac{|\Delta \vec{v}|}{\Delta t}$ , using the vector change in velocity. Over half a circle the velocity reverses direction, so $|\Delta \vec{v}| = 2v$ ; the time is the arc length over the speed, $\Delta t = \dfrac{\pi r}{v}$ . Hence the average acceleration over half a circle is $\dfrac{2v}{\pi r / v} = \dfrac{2v^2}{\pi r}$ .

Average acceleration over half a circle

\bar{a} = \dfrac{|\Delta \vec{v}|}{\Delta t} = \dfrac{2v}{\pi r / v} = \dfrac{2v^2}{\pi r}

$\Delta \vec{v}$ vector change in velocity (= 2v over half a circle)
$\Delta t$ time for the half circle (= πr/v)
vconstant speed

Worked example

A particle moves at constant speed v on a circle of radius R. Find its average acceleration over the time it covers half the circle.

Practice this conceptself-check · 3 quick reps

From the bank · past-year question

Example 3Kinematics and MotionHARD

A particle is moving in a circle of radius R with a constant speed v. Its average acceleration over the time when it moves over half the circle is :

[Q55 · Apr · 2023]

Average acceleration is not the instantaneous v²/r

v²/r is the instantaneous centripetal acceleration. The AVERAGE over half a circle uses |Δv|/Δt = 2v ÷ (πr/v) = 2v²/πr — a different expression. And over a full revolution the average acceleration is zero (Δv = 0), even though the instantaneous value is never zero.

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (2)

Centripetal acceleration — v²/r toward the centre
Centripetal acceleration
$a_c = \dfrac{v^2}{r}$
Average acceleration over part of a circle
Average acceleration over half a circle
$\bar{a} = \dfrac{|\Delta \vec{v}|}{\Delta t} = \dfrac{2v}{\pi r / v} = \dfrac{2v^2}{\pi r}$

Watch out for (3)

Constant speed is not constant velocity
→ Uniform circular motion — constant speed, changing velocity
Centripetal acceleration does not change the speed
→ Centripetal acceleration — v²/r toward the centre
Average acceleration is not the instantaneous v²/r
→ Average acceleration over part of a circle

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