Barrons AP Calculus - David Bock

How integration may be used to solve problems of curvilinear motion is illustrated in the
following examples.

BC ONLY

EXAMPLE 4

Suppose a projectile is launched from the origin at an angle of elevation α and initial velocity v 0.

Find the parametric equations for its flight path.

SOLUTION: We have the following initial conditions:

Position: x(0) = 0; y(0) = 0.

Velocity:

We start with equations representing acceleration due to gravity and integrate each twice,

determining the constants as shown:

If desired, t can be eliminated from this pair of equations to yield a parabola in rectangular

coordinates.

EXAMPLE 5

A particle P(x, y) moves along a curve so that

at any time t 0.

At t = 0, x = 1 and y = 0. Find the parametric equations of motion.

SOLUTION: Since we integrate to get and use

x(0) = 1 to find that C = 2. Therefore, and

Since y(0) = 0, this yields C ′ = 1, and so (2) becomes