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Many applications of calculus to engineering or science involve parametric curves or vector valued functions, including the examples listed below.

- The kinematics of mechanisms, like internal combustion engines.
- Population models involving more than one species.
- Chemical kinetics in systems with more than one chemical species.
- Forced or transient responses of electrical circuits.

For the applications we will deal with, we only need to consider
curvilinear motion of a point in the plane, That is, we have a
vector-valued function that gives the position at time
*t* of a moving point *P* in the plane. The velocity of this point is
given by the derivative and the acceleration is given
by the second derivative, . If the velocity,
, is not the zero vector, then it is clear from the
way it is defined that is a vector that is tangent to
the curve at the point . A simple example of
curvilinear motion is when the velocity is constant. That is, there is
a fixed vector such that

In many applications of
curvilinear motion, we need to know the magnitude of the velocity, or
the speed. This is easy to compute - just take the magnitude . If *f* and *g* are the component functions of
, then the speed is given by

If the speed is not zero for any value of *t* in the interval *I*,
then it is possible to define a unit vector, that is
tangent to the curve as follows.

Now suppose, to carry our analogy a little further, that you wanted to analyze the acceleration of a car as it traversed the road. On sections where the road is straight, acceleration depends only on whether the car is speeding up or slowing down. In a curve, however, there is an additional acceleration, called the centripetal acceleration, that is needed to keep the car on the road. The magnitude of this acceleration depends on the speed of the car and the sharpness of the curve. It turns out that you can quantify the sharpness of a curve with an intrinsic property of the curve called the curvature, usually denoted , defined by the following equation.

That is, the curvature is the magnitude of the rate of change of the tangent vector with respect to arc length. For example, the curvature of a straight line is zero and it can be shown that the curvature of a circle of radiusIf the curvature is never zero for a particular curve, then we can define another intrinsic property of curve, the unit normal vector by the following equation.

It can be shown that at each point on the curve the vector defined by this equation is a unit vector that is always perpendicular to the tangent vector
Computing these intrinsic properties of a curve is generally not an
easy task. The **Getting started** worksheet for this lab
describes commands from the `CalcP` package that simplify these
calculations and provides examples for you to work from.

- 1.
- Suppose that at time zero, flight 12 is at the point
(-100
*mi*,-100*mi*) at an altitude of 30,000 feet and traveling Northeast at 420 mph and that flight 33 is at the same altitude, but is traveling due west at a speed of 388 mph. At time zero, flight 33 is at the point (200*mi*, 30*mi*).- (a)
- What is the minimum distance between flights 12 and 33, and at what time does it occur?
- (b)
- Suppose that FAA regulations require that the closest approach of any two aircraft must not be less than 2000 feet. Can you change the speed of flight 33 so that this regulation is satisfied? What is your new speed, and what is the closest approach distance?

- 2.
- Consider the curve in the plane given by
*y*=*x*for and the two different parametrizations given below. In each case^{2}*t*is restricted to the interval .- (a)
- (b)

- (a)
- Describe the difference between these two different parametrizations.
- (b)
- Compute the unit tangent vector at the point
*x*=0,*y*=0 for each parametrization and explain your results. - (c)
- Compute the curvature and the unit normal vector at the point
*x*=0,*y*=0 for each parametrization and explain your results.

- 3.
- A hairpin turn on a roadway can be approximated as half of an
ellipse, including the half of major (longer) axis. If the major axis is 100
feet and the minor axis is 80 feet, what is the maximum (constant)
speed a car can go through the turn while keeping the centripetal
acceleration less than 0.3
*g*, where*g*is the acceleration due to gravity? - 4.
- Consider . What is the
normal vector at
*t*=0.01? At*t*=-0.01? What about at*t*=0? What goes wrong? - 5.
- For with , answer the following questions.
- (a)
- For which value of
*t*is the speed a maximum? - (b)
- For which value of
*t*is the curvature a maximum? - (c)
- For which value of
*t*is the tangential acceleration a maximum? - (d)
- For which value of
*t*is the normal acceleration a maximum?

3/23/1999