A motorboat starts at the origin and moves in the direction of the positive x-axis, pulling a waterskier along a curve C called a tractrix. See the figure below. The waterskier, initially located on the y-axis at the point (0, a),is pulled by a rope of constant length that is kept taut throughout the motion. At time  t > 0 the waterskier is at point P(x, y). Assume that

EXERCISE 2.2 An automobile follows a circular road whose radius is 50 m. Let x and y respectively denote the eastern and northern directions, with origin at the center of the circle. Suppose the vehicle starts from rest at x = 50 m heading north, and its speed depends on the distance s it travels according to v=0.5s-0.0025s², where s is measured in meters and v is in meters per second. It is known that the tires will begin to skid when the total acceleration of the vehicle is 0.6g. Where will the automobile be and how fast will it be going when it begins to skid? Describe the position in terms of the angle of the radial line relative to the x axis.

Consider the curve r(t) = (t² + 1)i + tj + (2t² + 3t)k. What is the velocity? What is the accelera- tion? Are they constant?

Consider the trajectory r(t) =   t2+1t2-vt+1. Find a vector n per pendicular to the curve describe by this trajectory at time t = v. Hint: Sketch a schematic figure showing axes, the origin, the curve described by this trajectory, by the position vector r(t) at time t=v, and the vector n to be found.   Answer: n = <___,___>