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.



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 d2θ/dt2. 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
14 Dec 1999