### 6th Putnam 1946

**Problem B6**

The particle P moves in the plane. At t = 0 it starts from the point A with velocity zero. It is next at rest at t = T, when its position is the point B. Its path from A to B is the arc of a circle center O. Prove that its acceleration at each point in the time interval [0, T] is non-zero, and that at some point in the interval its acceleration is directly towards the center O.

**Solution**

This is almost trivial. Take polar coordinates with origin O. Let the radius of the circular arc be k. The particle's radial acceleration is k (dθ/dt)^{2} towards O. Its tangential acceleration is k d^{2}θ/dt^{2}. Since the particle is not at rest between A and B, (dθ/dt) is non-zero, so the radial acceleration is non-zero (and hence its total acceleration).

(dθ/dt) is zero at A and at B, so its derivative (and hence the tangential acceleration) must be zero at some point between.

6th Putnam 1946

© John Scholes

jscholes@kalva.demon.co.uk

14 Dec 1999