The Child and the Toy

Let us suppose that a child is walking on the plane. His orbit is given by the two functions of time and .

Suppose now that the child is pulling some toy, to which he is connected by a rigid bar of length a. We are interested in computing the orbit of the toy when the child is walking around. Let be the position of the toy. Based on these relationships we can formulate the following equations:

1. The distance between then point and is always the length of the bar. Therefore

   

2. The toy is moving always in the direction of the bar. Therefore the difference vector of the two positions is some multiple of the velocity vector of the toy :

   

   with .

3. The absolute value of the velocity of the toy depends on the direction of the velocity vector of the child . Assume e.g. that the child is walking on a circle of radius a (length of the bar). In this special case the toy will stay in the center of the circle and not move at all.

   In general, the absolute value of the velocity of the toy is given by the absolute value of the projection of the velocity of the child onto the bar.