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:
Acceleration:
Finally, then,
x = (v 0 cos α)t; y = − gt 2 + (v 0 sin α)t.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
and At t = 0, x = 1 and y = 0. Find the parametric equations of motion.
SOLUTION: Since dt, we integrate to get 2 = 2t + C, and use x(0) = 1
to find that C = 2. Therefore, = t + 1 and
x = (t + 1)^2.
(1)