Motion under gravity

Lets consider a simple example, a body is thrown vertically into the air at a velocity of 10 m/s. If the acceleration due to gravity is 10 m/s² and air resistance can be ignored, what is the maximum height attained  by the body ?

Assuming the upward direction to be positive, the equation to be solved is y" = -10, with the initial conditions y = 0 and y' = 10. The plot of the solution is as below, from which we see that the body reaches a maximum height of 5 m.

Now, lets consider a situation where the air resistance cannot be ignored, for example in the case of a falling parachutist or a body falling in a viscous fluid. In this case, there is a fluid resistance that which acts against the motion and is proportional to the velocity.

We can use the ODE solver tool to model such situations and we find that the velocity of the body that falls under gravity tapers off to a terminal velocity. The graph below is the plot of velocity (y'), which is obtained as the solution of the equation y" = 2 - y',  initial conditions are y = 0 and y' = 0.