Honors Calculus Mat 119, Spg 1999

Task 1: Curve of Least Pursuit

In this task we will explore a differential equation problem.

Prerequisites: Techniques of integration and separation of variables method.

Objectives: To show an elementary use of calculus.

Problem:

A Fighter pilot notices at the same elevation an enemy bomber which is a miles away, cruising on a straight line at constant speed k (mph). The Fighter steers towards the enemy and maintains course on him at a speed nk (mph). Assuming no evasive action on the part of the enemy, find the path of the fighter.

Part 1: Due Wed, March 17

Task T1: Let (x, y) denote the coordinates of the Fighter pilot. x and y are functions of t (time). So, the velocity of the Fighter pilot is the vector v(t) = (dx/dt, dy/dt). a. What is the position of the Bomber at time t? b. Find the vector v(t), that is, where is this vector pointing to? The

expression for the vector v(t) will be in terms of x, y, a, k, and t.

Task T2: From task T1, you have expressions for dx/dt and dy/dt. Find dy/dx. Geometrically, what does dy/dx represent?

Part 2: Due Fri, March 19

Task T3: The speed of the Fighter pilot is the length of its velocity vector. That is speed= |v(t)|. Use this information to show

nk = \frac{dx}{dt}\sqrt{1 + (\frac{dy}{dx})^{2}}.

Recall that v(t) = (dx/dt, dy/dt).

Task T4: In T2 you found that

\frac{dy}{dx} = \frac{kt − y}{a − x} .

We would like to solve for the parameter k so that our final solution for y will not depend on k. Write

(a − x)\frac{dy}{dx} = kt − y

To solve for k, take derivative with respect to x in both sides of the above equation. To make notation easier, write y′ = dy/dx.

Part 3: Due Mon, March 29

Task T5: First attempt to solve for y. From T3 we had that nk = (dx/dt)\sqrt{1 + (y′)^{2}}, and from T4 we found that k(dt/dx) = (a − x)y′′. a. Put T3 and T4 together to show

n(a − x)y′′ = \sqrt{1 + (y′)^{2}}.

b. Rewrite this equation making the substitution u = y′. c. What is the initial value information? That is, if x = 0 what is the

value of u? You may have to look at what you got in T2. d. Use the separation of variables method in order to attempt to solve

u in terms of x. e. Now you can solve fot u using the definition of sinh. Make sure to

compute the constant.

Part 4: Due Tue, April 9

Task T6: Now we are finally ready to solve for y. Use the information in task T6 to a. solve for y when n = 1. b. solve for y when n ≠ 1. Remember that y = 0 when x = 0.

Recall that in T6 we got:

u = \frac{1}{2} [(\frac{a}{a − x})^{1/n} − (\frac{a − x}{a})^{1/n}] .

Task T7: Plot y(x) when n = 1 and when n = 2.