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Projectile with Air Resistance

Compare ideal and real projectile trajectories side by side. Uses RK4 numerical integration for the drag trajectory. Calculates terminal velocity, range reduction, and time of flight difference.

Trajectory Comparison

Ideal (no drag)With drag

Parameters

Launch Speed40 m/s
Launch Angle45 °
Mass0.145 kg
Drag Coefficient (Cd)0.47
Cross-section Area (A)0.0043 m²
Air Density (ρ)1.225 kg/m³
Initial Height0 m

Results

Ideal (no drag)With drag
Range163.20 m84.30 m
Max Height40.77 m27.36 m
Time of Flight5.77 s4.70 s
Range Reduction
48.3%
Terminal Velocity
34.1 m/s
Time Difference
1.07 s
Height Reduction
32.9%
Terminal Velocity
vt=2mgρCdA=34.1 m/sv_t = \sqrt{\dfrac{2mg}{\rho C_d A}} = 34.1 \text{ m/s}
Terminal velocity reached during flight

Reference Guide

Air Resistance

Air resistance (aerodynamic drag) opposes the motion of an object through a fluid. Unlike gravity, drag depends on both speed and direction, making the equations of motion nonlinear and requiring numerical integration.

Two drag regimes exist. At low Reynolds numbers (Re < 1), Stokes drag applies and force scales with velocity. For projectiles at everyday speeds, the quadratic (Newton) drag law dominates where drag scales with v².

The result is an asymmetric trajectory: the projectile rises steeply and falls more steeply than the ideal parabola, landing closer to the launch point.

Drag Force

The quadratic drag force magnitude is:

Fd=12 ρ v2 Cd AF_d = \tfrac{1}{2}\,\rho\, v^2\, C_d\, A

where ρ is air density (kg/m³), v is speed (m/s), Cd is the dimensionless drag coefficient, and A is cross-sectional area (m²). The force direction is always opposite to velocity.

Typical Cd values: sphere ~0.47, streamlined body ~0.04, cube ~1.05, skydiver prone ~0.70. The acceleration components are:

ax=−Fdmvxv,ay=−g−Fdmvyva_x = -\frac{F_d}{m}\frac{v_x}{v},\quad a_y = -g - \frac{F_d}{m}\frac{v_y}{v}

Terminal Velocity

Terminal velocity is reached when drag exactly balances gravity on a falling object. Setting Fd = mg and solving for v:

vt=2mgρ Cd Av_t = \sqrt{\frac{2mg}{\rho\, C_d\, A}}

Heavier objects have higher terminal velocities. Objects with large Cd or large area reach terminal velocity at lower speeds. A skydiver in freefall (~55 m/s prone, ~90 m/s head-down) vs. a baseball (~42 m/s).

Once terminal velocity is reached during a trajectory, vertical acceleration drops to zero and the projectile descends at constant speed.

Range Comparison

The ideal range (no drag) for h0 = 0 is the analytic result:

Rideal=v02sin⁡2θgR_{ideal} = \frac{v_0^2 \sin 2\theta}{g}

Maximum ideal range occurs at 45°. With drag, the optimal angle is always below 45° because slowing down reduces the horizontal component more than a shallower angle would.

The numerical range with drag must be found by integrating the full equations of motion (RK4). Range reduction can exceed 50% for light objects at high speeds, while dense objects like cannonballs experience much smaller reductions.