How to do Air Friction on a Sky Diver

Here is a case where air friction acts on a falling object, such as a sky diver. This is one of the trickier math problems we will do in physics, as it involves calculus (anything with air friction will, since it is a non-constant force: f = -kv). We specifically want to solve for the velocity as a function of time for the sky diver. Check this out to get a feel for how Newton's 2nd law sets up the equation, and then we do almost all algebra with a step of calculus to solve for velocity. Note that terminal velocity is the speed you reach when air friction matches the strength of gravity, and the person falls with a constant speed at that point. Also note that we do a very simplified model of air friction. Other factors we do not worry about here include the shape of the object, air density that varies with altitude, wind, air temperature that varies, the material of the object, the gaseous composition, and so on (for us, all this information is contained in the constant, k).

Spring Problems and the use of Energy

Springs provide a nice example of NON-constant force. We know Hooke's law, F = -kx, where x is the displacement of the spring from its equilibrium position. Keep in mind the - sign is for direction: the force always tries to bring a spring back to equilibrium, which is in the opposite direction relative to the direction of the displacement (a point of stable equilibrium). Here are examples of using energy conservation to solve spring problems.