Effect of Air Resistance on Projectile

What air resistance does

In all standard projectile formulas, we assume no air resistance. In reality, air exerts a drag force on the moving projectile — always opposing the direction of motion.

This drag force has two effects:

• It opposes horizontal motion → horizontal velocity decreases throughout flight
• It opposes vertical motion → reduces upward velocity faster on the way up, and limits downward velocity on the way down

How every quantity changes

Range

$R_{actual} < \frac{u^2\sin 2\theta}{g}$

Air resistance continuously reduces the horizontal velocity. Since horizontal velocity is no longer constant, the body covers less horizontal distance. The actual range is always less than the vacuum range.

Maximum height

$H_{actual} < \frac{u^2\sin^2\theta}{2g}$

On the way up, the drag force has a downward component (opposing upward motion). This adds to gravity, decelerating the projectile faster. It reaches a lower maximum height than in vacuum.

Time of flight

The time of flight is not simply reduced — it splits into two unequal halves:

• Time of ascent is shorter than in vacuum — drag adds to gravity on the way up.
• Time of descent is longer than in vacuum — drag opposes gravity on the way down, slowing the fall.

Overall, the time of flight may be slightly less than, equal to, or greater than the vacuum value depending on the strength of drag. For most realistic cases with moderate air resistance, the total time is slightly reduced.

Speed at landing

In vacuum, the landing speed equals the launch speed. With air resistance:

$|\vec{v}_{landing}| < |\vec{v}_{launch}| = u$

The projectile loses kinetic energy to the drag force throughout its flight. It arrives at the ground with less speed than it was launched with.

Symmetry

In vacuum, the trajectory is a perfect parabola — symmetric about the vertical through the peak. With air resistance, the trajectory is asymmetric:

• The ascending part is less steep (wider)
• The descending part is steeper (narrower)

The peak is shifted toward the landing side — closer horizontally to where it lands than to where it was launched.

The optimal angle shifts below 45°

In vacuum, maximum range is at $\theta = 45°$. With air resistance, the optimal angle is less than 45° — typically around $30°$–$40°$ depending on the drag.

The reason: at higher angles, the projectile moves more slowly (air resistance reduces speed), so the advantage of extra time in the air is outweighed by the loss of horizontal velocity. A flatter trajectory with more horizontal velocity gives better range.

Terminal velocity on descent

For very long falls (or very light/large projectiles), the drag force eventually balances gravity on the way down — the projectile reaches terminal velocity and no longer accelerates. The descent slows to a constant speed. This does not occur in standard projectile problems but is important in understanding falling objects in real air.

Summary table