284 CHAPTER 15. MOTION ON A SPACE CURVE

19. Consider the curve obtained from the graph of y = f (x). Find a formula for thecurvature.

20. Consider the curve in the plane y = ex. Find the point on this curve at which thecurvature is a maximum.

21. An object moves along the x axis toward (0,0) and then along the curve y = x2 inthe direction of increasing x at constant speed. Is the force acting on the object acontinuous function? Explain. Is there any physically reasonable way to make thisforce continuous by relaxing the requirement that the object move at constant speed?If the curve were part of a railroad track, what would happen at the point wherex = 0?

22. An object of mass m moving over a space curve is acted on by a force F. Show thework done by this force equals maT (length of the curve). In other words, it is onlythe tangential component of the force which does work.

23. The edge of an elliptical skating rink represented in the following picture has a lightat its left end and satisfies the equation x2

900 +y2

256 = 1. (Distances measured in yards.)

(x,y)z

L

T

A hockey puck slides from the point T towards the center of the rink at the rate of 2yards per second. What is the speed of its shadow along the wall when z = 8? Hint:You need to find

√x′2 + y′2 at the instant described.