17.1 Modeling Vertical Motion

We observed, back in Chapter 4 that the rules for differentiating rational functions could all be deduced from one master rule: each occurrence of the variable in the function to be differentiated can be replaced by \(1\), ignoring the others, and the derivative is the sum of the results. This statement represents the fact that the derivative is the slope of the linear approximation to the function and is linear in the variable, and linear contributions can be evaluated one by one and added.

Essentially this same property implies that in constructing models of the behavior of derivatives for real phenomena, the effects on the derivative from different sources can be entered separately, one at a time, ignoring the others, and the total effect will be the sum of these.

Consider now vertical motion of an object. Newton observed that, if an object is left alone, it will continue doing what it was doing, so that its speed will stay constant. How that speed changes, which can be described by its derivative, is then proportional to what he called the "forces" compelling it to change.

Apples fall from trees, and with increasing speed as they fall. He attributed that behavior to the force of gravity, and his model for that force is that objects experience a constant negative pull of gravity toward the earth, while on its surface.

It is obvious that heavier objects require more force to move them. His model therefore was that the weight (mass) \(m\) of the object multiplied by the second derivative of its height \(h(t)\) is given by the force of gravity acting on it. Noticing that objects fall at rates independent of their weight, his model for that force was \(mg\), with \(g\) a universal constant.

His model for falling objects was then

\[mh''(t) = -mg\]

We can solve this equation. The velocity \(h'(t)\) must have derivative \(-g\), which is a constant. The general solution to this is \(h'(t) = h'(t_0) - g(t-t_0)\). This tells us that the derivative of \(h(t)\) is a linear function of \(t\), and that means that \(h(t)\) is a quadratic function:

\[h(t) = h(t_0) + h'(t_0)(t-t_0) - g\frac{(t-t_0)^2}{2}\]

Now let is consider air resistance. Objects have air resistance that depends on their shape and size. For any object there is no air resistance when it is at rest, and so the simplest model for it is that the force of the air on it is linear in its speed and in the opposite direction to it: say \(-ch'(t)\).

The equation for \(h''(t)\) then becomes

\[h''(t) = -g - \frac{c}{m}h'(t)\]

Notice that the right side of this equation is \(0\) when \(h'(t) = -\frac{gm}{c}\). This means that a falling object starting at rest, will fall faster and faster until its downward speed reaches this value, at which time its speed will become constant. Thus the object (imagine a person with a parachute) will achieve this ‘terminal velocity’ instead of falling ever faster until it hits something.

We can solve this equation by defining \(y\) to be \(h'(t) + \frac{gm}{c}\); \(y\) has the same derivative as \(h'(t)\) so that the equation for it is \(y' = -\frac{c}{m}y\). The solution to this equation is \(y(t) = y(t_0)e^{-\frac{c}{m}(t-t_0)}\), which means that a falling object, according to this model, approaches its terminal velocity exponentially fast, with exponent \(-\frac{c}{m}t\).

You will notice that while in the model for \(h''(t)\) the contributions from gravity and air resistance are separate added terms, these contributions mix completely in the solution for \(h'(t)\).

There are more interesting problems of this sort involving the behavior of objects in ordinary three dimensional space.

Newton invented calculus to solve the equations that derived from his models. In particular he applied it to describing the motion of planets which are attracted to one another and to the sun by the force of gravity with each pair separately attracting each other with a force proportional to the masses of both divided by the square of their distance. Again, the forces on any planet are the sum of those from all the others. To a first approximation the attraction from the sun is dominant, and he was able to solve the equations for planetary motion of one planet about the sun. The solutions are that the orbit is an ellipse with the Sun at one of its foci. Calculus with many variables allows formulation and solution of these equations.