Preliminaries. The following mathematical models for projectiles are considered.
No resistance yields
.

Resistance proportional to velocity yields
.

Resistance proportional to the square of the velocity yields
, for the ascent, and
, for the descent.

Computer Lab Work.

Example 1. An arrow is shot upward from the origin with an initial velocity of 300 ft/sec.
Assume that there is no air resistance. Find the velocity and position as a function of time.
Find the ascent time, the descent time, maximum height, and the impact velocity.

Notice that the maximum altitude will occur when the time is near t = 9,
and the arrow will hit the ground when the time is near t = 18.

Example 2. An arrow is shot upward from the origin with an initial velocity of 300 ft/sec.
Assume that air resistance is proportional to the velocity, e.g. .
Find the velocity and position as a function of time, and plot the position function.
Find the ascent time, the descent time, maximum height, and the impact velocity.

Notice that the maximum altitude will occur when the time is near t = 8,
and the arrow will hit the ground when the time is near t = 17.

Example 3. An arrow is shot upward from the origin with an initial velocity of 300 ft/sec.
Assume that air resistance is proportional to the square of the velocity, e.g. .
Find the velocity and position as a function of time, and plot the position function.
Find the ascent time, the descent time, maximum height, and the impact velocity.

Notice that the maximum altitude will occur when the time is near t = 8,
and the arrow will hit the ground when the time is near t = 15.

Remark. No matter how much you like the above model, it isn't right.
With air resistance the descent time must be greater than the ascent time !
The D. E. for the descent must have the sign of the term with positive.

Notice that the maximum altitude will occur when the time is near t = 8,
and the arrow will hit the ground when the time is near t = 16.

(c) John H. Mathews, 1998