Changing Direction of Velocity
When an object moves along a curved path, its velocity is constantly changing direction — even if its speed remains perfectly constant. This is the essence of curvilinear motion. Velocity is a vector quantity, meaning it has both magnitude (speed) and direction. On a curved path, the direction of the velocity vector is always tangent to the curve at each point. Because direction is changing continuously, the velocity vector itself is changing, which means the object is accelerating — even at constant speed. This surprises many students who associate acceleration only with changes in speed. A classic example is a car travelling around a roundabout at a steady 30 km/h. The speedometer needle never moves, yet the car is accelerating because its direction changes every instant. The same principle applies to the Moon orbiting Earth, a ball on a string being swung in a circle, or a roller-coaster navigating a bend. In all these cases, something must be causing the change in the velocity vector's direction — a net force acting towards the centre of curvature, which we call the centripetal force. Without it, the object would fly off in a straight line, as Newton's first law demands. Understanding curvilinear motion requires a solid grasp of vectors. The velocity vector at any instant points along the tangent to the path. As the object progresses along the curve, this tangent direction rotates, so the velocity vector rotates with it. The rate at which the velocity vector changes — both in direction and potentially in magnitude — defines the acceleration vector, which in general points towards the inside of the curve.
- Velocity is always tangent to the path
- Changing direction means changing velocity
- The acceleration vector points inward
- Radius of curvature governs the acceleration
Velocity is always tangent to the path
At every point on a curved path, the instantaneous velocity vector points along the tangent to the curve at that point. Think of mud flying off a spinning bicycle tyre — it always leaves in the direction the tyre surface was moving at that instant, which is tangential. If you drew the path as a curve on paper and placed a ruler just touching it at one point without crossing it, the ruler shows the direction of velocity there.
Changing direction means changing velocity
Even if the speed (magnitude of velocity) is constant, the velocity vector is changing because its direction changes. Mathematically, if v₁ and v₂ are velocity vectors at two nearby points on the path, the change Δv = v₂ − v₁ is a non-zero vector pointing roughly towards the inside of the curve. This is a vector subtraction: place the two arrows tail-to-tail and the difference arrow closes the triangle. As the time interval Δt shrinks, Δv/Δt gives the instantaneous acceleration vector.
The acceleration vector points inward
For uniform circular motion (constant speed on a circle), the acceleration vector always points directly towards the centre of the circle. This is called centripetal acceleration, from the Latin meaning 'centre-seeking'. For non-uniform curvilinear motion, the acceleration has two components: a centripetal (radial) component directed towards the centre of curvature, responsible for changing direction, and a tangential component along the path, responsible for changing speed. Both components together describe how the velocity vector evolves.
Radius of curvature governs the acceleration
The sharper the bend (smaller radius of curvature r) and the greater the speed v, the larger the centripetal acceleration required. This is why a car must brake before a tight corner — at high speed on a sharp bend, the required centripetal force may exceed what the tyres can provide. The formula aᶜ = v²/r captures this relationship precisely: doubling the speed quadruples the required centripetal acceleration for the same bend.
The Key Equation
- a_c — Centripetal acceleration (m s⁻²): The component of acceleration directed towards the centre of curvature, responsible for changing the direction of the velocity vector.
- v — Speed (m s⁻¹): The magnitude of the velocity vector — how fast the object is moving along the curved path at that instant.
- r — Radius of curvature (m): The radius of the circular arc that best fits the curve at that point. A tighter bend has a smaller radius of curvature.
Worked Example
Problem. A cyclist travels around a circular roundabout of radius 12 m at a constant speed of 6 m s⁻¹. (a) Calculate the centripetal acceleration of the cyclist. (b) In what direction does this acceleration act at every point on the roundabout?
Answer. The centripetal acceleration is 3 m s⁻², directed towards the centre of the roundabout at every instant.
Origins · 1 minChristiaan Huygens and how this was discovered
Huygens was the first to derive the correct formula for centripetal acceleration, publishing his results in 'De Vi Centrifuga' (written c. 1659, published 1703). He showed rigorously that for circular motion at constant speed, the inward acceleration equals v²/r. This was a pivotal result in the development of classical mechanics, predating Newton's published work on the topic and directly influencing Newton's own formulation of the laws of motion and universal gravitation. Huygens also invented the pendulum clock in 1656, dramatically improving timekeeping accuracy, and was the first to correctly describe the rings of Saturn — identifying them as a flat disc surrounding the planet.
Related Concepts
- Uniform Circular Motion — Uniform circular motion is the special case of curvilinear motion where the radius is constant and speed is unchanged, making it the clearest demonstration of velocity direction change and centripetal acceleration.
- Velocity and Speed — Understanding that velocity is a vector (with direction) while speed is a scalar (magnitude only) is essential to grasping why direction change alone constitutes acceleration in curvilinear motion.
- Acceleration — Acceleration is defined as the rate of change of velocity. In curvilinear motion, acceleration arises from direction change, making a deep understanding of the vector definition of acceleration necessary.
- Projectile Motion — Projectile motion is another form of curvilinear motion where both speed and direction change simultaneously, providing a practical context for applying the principles of changing velocity direction.