The Physics

Understanding projectile motion and drag.

Ideal Projectile Motion

Fundamentally, projectile motion studies the motion of objects being influenced only by the force of gravity, and can be governed by the three kinematic equations:

\( v = v_i + at \)
\( \Delta x = v_i t + \tfrac{1}{2}at^2 \)
\( v^2 = v_i^2 + 2a\Delta x \)

Projectile motion defines a constant horizontal velocity component and a vertical velocity component accelerating at the linear rate of ambient gravity \(-g\), independent of mass:

Vertical
\( F_{net,y} = -F_g = -mg \)
\( a_y = -g \)
Horizontal
\( F_{net,x} = 0 \)
\( a_x = 0 \)

The Drag Force

However, these relationships do not work in practice due to the nonlinear force of drag:

\( F_d = \tfrac{1}{2} \rho C_d A v^2 \)

This means projectiles stray from their parabolic trajectories, especially when experiencing higher drag forces due to larger cross-sectional areas, higher drag coefficients, faster velocities, or denser fluids.

Motion with Drag

When drag is introduced, the equations of motion become more complex:

Vertical
\( F_{net,y} = -\left(mg + \tfrac{1}{2}\rho C_d A v_y^2\right) \)
\( a_y = -\dfrac{mg + \tfrac{1}{2}\rho C_d A v_y^2}{m} \)
Horizontal
\( F_{net,x} = -\tfrac{1}{2}\rho C_d A v_x^2 \)
\( a_x = -\dfrac{\tfrac{1}{2}\rho C_d A v_x^2}{m} \)

Note that mass no longer cancels, due to the added drag force term. Acceleration is also now variable, as it depends on the instantaneous velocity.

How the PMADS Works

This is the fundamental function of the PMADS - it models velocity, position, and time values to compute instantaneous forces acting on the projectile and its acceleration, to correctly map out its trajectory. Using this simulation, you can graphically understand how drag influences projectile motion. Use the density presets to see how denser fluids cause higher drag, and how the vacuum preset shows perfect projectile motion.