ACCELERATED MOTION OF A FALLING OBJECT

by Donald F. Elger

Situation

Consider an object of mass m released from rest and allowed to free fall.  Assume aerodynamic drag acts on the object with a force given by BV², where B is a constant that contains the coefficient of drag, projected area, and air density.

Goal

Find an equation for the speed and position of the object. 

Method

1. Apply Newton's 2nd law of motion to yield  W - BV² = m(dV/dt)
2. Separate variables and integrate using definite integrals.  The limits of integration are 0 to t (for time) and 0 to V(t) (for speed)
3. Use the hyperbolic sine and cosine (that is, sinh and cosh) to clean up the algebra
4. Integrate ds = Vdt using definite integrals.

Result

The final equations (written using MathCAD notations are):  

where v(t) is object speed, t is time, V_t is terminal velocity, g is the gravitation constant, and s(t) is object position.  These equation may be  programmed in MathCAD version 7.

Review

Solution and the MathCAD program have been checked by using MathCAD to  numerically integrate Newton's second law

When t becomes large, the hyperbolic tangent term in the velocity equation approaches 1.0 and speed approaches terminal velocity.