Projectile with Air Drag

The plots show projectile motion with air resistance (red) compared with the same motion neglecting air resistance (blue). The projectile is launched at an angle with initial velocity . The force due to air resistance is assumed to be proportional to the magnitude of the velocity, acting in the opposite direction.
A significant decrease in the maximum horizontal range is observed when the drag force becomes large. When this value is large, the terminal velocity (the maximum velocity for a falling object) is reduced. Independent of the initial value of the angle, the projectile ends up falling vertically if it stays in the air long enough before it hits the ground.
A more accurate model of air drag considers another contribution proportional to the square of the velocity, but is more difficult to treat analytically.

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DETAILS

The equations of motion for the and directions are given by
,
where increases upward and is a positive constant. The terminal velocity is given by , so the equations can be simplified to
.
For a projectile launched at an angle , and .
Integration of the equations of motion gives
,
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