Centripetal Acceleration (a = v²/r)

An object going round a circle at perfectly constant speed is nevertheless accelerating, and that sentence is the first thing to make peace with. Acceleration is the rate of change of velocity, and velocity is a vector with a direction as well as a size. Something moving in a circle is having its direction changed continuously, so its velocity is changing continuously, so it is accelerating — even though a speedometer strapped to it would never move. The acceleration points at the centre of the circle, and its size is $a_c = v^2/r$.

A 1200 kg car rounding a 50 m curve at 20 m/s — 72 km/h — experiences $a_c = 20^2/50 = 8$ m/s², about 0.82 g. That is close to the limit of what a good tyre on dry pavement can supply, which is why that corner at that speed feels like it is asking a real question. Take the same corner at 30 m/s and the demand rises to 18 m/s², about 1.8 g, and no ordinary road tyre will hold it.

The derivation is short enough to be worth carrying. Over a small time $\Delta t$, the position vector sweeps through an angle $\Delta\theta = v\Delta t/r$. The velocity vector, always at right angles to the position vector, must rotate through exactly the same angle, and rotating a vector of length $v$ through a small angle changes it by $|\Delta v| = v\Delta\theta$. Put the two together: $|\Delta v| = v^2\Delta t/r$, so $a = v^2/r$. Christiaan Huygens published this result in 1673, and it is what let Newton check the inverse-square law against the Moon's orbit.

There is no outward force, and this is the single most persistent misconception in mechanics. In the ground frame nothing pushes you outward in a turning car. What happens is that your body would continue in a straight line, the car turns underneath you, and the door pushes you inward. The sensation of being flung out is your inertia, not a force. "Centrifugal force" is a bookkeeping term that appears only when you insist on doing the physics in the rotating frame, where it is added artificially so Newton's laws balance. It is a real effect and a useful device; it is not a force in an inertial frame, and there is no third-law partner to it. A more mundane error costs just as much: the $r$ is a radius, not a diameter. A component described as "600 mm diameter" has $r = 0.3$ m, and entering 0.6 halves the answer. Last, this is only the component of acceleration perpendicular to the motion. If the object is also speeding up or slowing down, there is a tangential component too, and the total acceleration is the vector sum of the two.