Free Fall with Air Resistance Calculator
Find falling speed and distance accounting for air resistance.
With drag, a falling object speeds up toward a terminal velocity rather than forever.
How the Math Works
The formula v = v_t·tanh(g·t/v_t) models an object's velocity during free fall with air resistance, where v_t is terminal velocity, g is gravitational acceleration, and t is time. Terminal velocity occurs when air resistance balances gravitational force, preventing further acceleration. The hyperbolic tangent function (tanh) arises from solving the differential equation dv/dt = g - (k/m)v, where k represents air resistance proportionality. As time increases, tanh approaches 1, causing velocity to asymptotically approach v_t, reflecting the physical reality that objects cannot exceed terminal velocity regardless of fall duration.
Practical Applications
This calculator is essential in physics experiments, engineering, and safety assessments where precise motion predictions matter. For instance, designing parachutes or impact barriers requires knowing how quickly objects reach terminal velocity to ensure safety margins. Sports scientists use it to analyze diving or ski jumping trajectories, while aerospace engineers apply it to model satellite or probe deceleration in planetary atmospheres. It also helps in forensic science to estimate falling object speeds in accident reconstructions by accounting for real-world drag forces rather than idealized vacuum conditions.
Day-to-Day Use
Understanding this calculation helps explain everyday phenomena, such as why raindrops don't fall at extreme speeds or why skydivers deploy parachutes at specific altitudes. It also aids in practical scenarios like estimating safe ladder heights when working on roofs or assessing risks for construction workers at great heights. Additionally, it provides intuitive insights into why bullets or coins dropped from tall buildings don't reach lethal velocities, debunking common myths while reinforcing fundamental physics principles in daily life.
Worked example
v_t 55 m/s after 5 s → about 41.7 m/s, having fallen ~125 m.
FAQ
What sets terminal velocity?
The balance of gravity and drag, depending on mass, area and drag coefficient.