Irodov Problem 1.36

Let us first understand what is happening in the problem. A particle moves along some arbitrary path (shown by the dotted line - imagine a roller coaster ride for example). It moves in such a way that its horizontal acceleration is always fixed and equal to a. In other words if the the particle is moving along a direction
then, the acceleration of the particle along the direction of its motion (along the tangential direction) is given by (the component of a along the tangential direction).

If the particle moves an infinitesimally small distance along the curved path ds, the distance it moves forward along the x-axis dx is given by,



This is clearly shown in the figure.

If the tangential velocity of the particle at any instant is given by , then we have,

Irodov Problem 1.35

(a) Actually this is almost identical to the previous problem since,






Now we can find the trajectory of the particle as,



(b) We can solve this problem both mathematically or using the expression for radial acceleration.

Using Mathematical definition for radius of curvature:
We know that the radius of curvature is given by, here x' and x'' represent the first and second derivatives with respect to time and the same for y.

(why? see here for the explanation).



From (1a,b) we have,











Using radial acceleration () :
The total acceleration of the particle is given by ab, as seen in above. The tangential acceleration of the particle can be determined by projecting the acceleration vector along the velocity vector to give,







The acceleration of the particle in the normal direction can be determined as,






The tangential velocity of the particle is same as the magnitude of the velocity vector given by |v|. Thus the radial acceleration is also given by,




This implies that,

Irodov Problem 1.34

a) From the definition of the problem we have,







From (1a,b) we can deduce,



b) From (1a,b) we again have,








Hence, we have,






The component of w along v is given by,






The component of w normal to v can be obtained as,