Taking the exponential of both sides givesĪnd solving for the time yields the time of impact from y peak: If we substitute the peak height y peak into the distance equation above so that it has reached the surface from which it was launched, we obtain the relationship Time of impact upon falling from peak height The constant of integration C=0 because cosh(0)=1 and ln(1)=0, so the result of the integration is The distance of freefall y as a function of time may be obtained by integrating the velocity expression above. The nature of the motion is such that the speed is essentially at its terminal velocity v t after a few characteristic times. If the falling object was released from rest at time t=0, the velocity expression becomes: The motion equation can then be solved for the velocity v: Which expresses the fall time t in terms of the characteristic time for the motion Which expresses the force in terms of the terminal velocity v t: A projectile is shot upward from the surface of Earth with an initial. The differential equation for the motion is use the position function s(t) - 4.9t2 + v0t + s0 for free-falling objects. (Free fall with air resistance) A projectile falling under gravity and slowed by air resistance proportional to its speed has position satisfying (frac. The expressions will be developed for the two forms of air drag which will be used for trajectories:Īlthough the first steps will be done with just the form -cv 2 for simplicity. The downward direction will be taken as positive, and the velocity as a function of time is the object of the calculation. Freefall Velocity with Quadratic Drag Freefall Velocity with Quadratic DragĪ freely falling object will be presumed to experience an air resistance force proportional to the square of its speed.