Exercises

1. Give a vector valued function whose graphical representation is a helix of radius 2 and whose axis is the straight line x=0, y=3.

2. Consider the two-dimensional position vector

   

   where b is a positive constant.

   1. Show that the trajectory, or image curve, of is the circle of radius 3 centered at the point . That is, show that the component functions and of satisfy the equation

      

      for all values of t. Explain why this means that the image curve is the circle in question.

   2. Show that the speed is constant and explain why this is so. Then find a value of the constant b that gives a speed of 18.
   3. Show that is perpendicular to .
3. Suppose particle 1 is going around a circle and its position is given by . Particle 2, on the other hand, is going back and forth on the line y=ax with position function , where a is a constant.
   1. If a=3, find the value of t, , for which the two particles collide.

   2. Is it possible to find a positive real number a for which the two particles do not collide for any value of t, ? Explain your answer.

4. In class, we derived the equations for the motion of a projectile. Now suppose that a projectile is to be fired from the origin up a hill that slopes uniformly at a angle. Find the angle , measured from the horizontal, that the projectile should be fired at to maximize the distance, measured along the hill, the projectile travels before it hits the hill.