Free Fall with Air Resistance (Speed and Distance from Time)

Compute falling speed and distance with air resistance from mass, terminal speed, and elapsed time. Shows the fraction of terminal speed reached and the difference from a vacuum.

Input

Compute falling speed and distance with air resistance from mass, terminal speed, and elapsed time.

Mass

Mass of the falling object

Terminal speed

m/s

Maximum falling speed where drag balances gravity

Elapsed time

Time since the fall began

Uses standard gravity g = 9.80665 m/s².

Result

Falling speed at that time

39.16773m/s

Distance 109.125139 m and 141.003827 km/h

Terminal speed

55 m/s

Fraction of terminal speed

71.214054 percent

Speed difference from vacuum

9.86552 m/s

Distance difference from vacuum

13.457986 m

In a vacuum (no drag), the speed would be 49.03325 m/s and the distance 122.583125 m.

Speed v = v_t tanh(g t / v_t), distance d = (v_t² / g) ln(cosh(g t / v_t)). Assumes quadratic drag.

How it works

Assumes quadratic drag (proportional to the square of speed) and takes the terminal speed v_t as a direct input.
Speed at time t is v = v_t tanh(g t / v_t), and distance is d = (v_t² / g) ln(cosh(g t / v_t)).
Uses standard gravity g = 9.80665 m/s².
Assumes an object dropped from rest, falling vertically downward.
The terminal speed v_t can be obtained from mass, air density, drag coefficient, and cross-sectional area via v_t = sqrt(2 m g / (rho Cd A)).
The difference from vacuum free fall (no drag) is shown for reference.

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