Problem

Projectile-motion-29-638.jpg

Assume the air resistance is proportional to the projectile's velocity. Determine the trajectory of a projectile fired at origin, and with initial velocity

.

(1) Determine the displacement vector.

(2) Let there be no motion in the y-axis. Determine .

Solution

Part (1)

To model the forces of air resistance, consider the net force in three dimensions

.

Divide the mass on both sides of the equation. Define , and rewrite the vector quantities in terms of the coordinates :

.

To obtain the velocity vector, integrate the acceleration with respect to time. Use the initial condition to solve for the constants.

To obtain the displacement vector, again, integrate the velocity with respect to time. Use the initial condition to solve for the constants.

Part (2)

Since , it follows that

.

Therefore

.
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