Question

A particle moves along the plane trajectory y (x) with velocity v whose modulus is constant. Find the acceleration of the particle at the point x = 0 and the curvature radius of the trajectory at that point if the trajectory has the form 

(a) of a parabola y = ax²; 

(b) of an ellipse (x/a)² +(y/b)² = 1; a and b are constants here.

Answer 1

(a) Let us differentiate twice the path equation y(x) with respect to time.

Since the particle moves uniformly, its acceleration at all point of the path is normal and at the point x=0 it coincides with the direction of derivative d²y/dt². Keeping in mind 

(b) Differentiating the equation of the trajectory with respect to time we see that 

is normal to the velocity vector

which, of course, is along the tangent. Thus the former vector is along the normal and the normal component of acceleration is clearly