PHYSICAL PROBLEMS
In physics, chemistry, mechanics, and other sciences there are many problems in which the behavior of distance, of time, of speed, heat, volume, electrical current, weight, acceleration, pressure, and many other physical quantities are related to each other. Examples of such problems are:
Fig. 6. Paths of a shot from a gun, trajectories.
What are the various angles to which a gun should be raised in order that it may shoot various distances? ([See Fig. 6.]) (The paths of a shot from a gun are called trajectories.)
If a plane flies in a direction always at the same angle from the north, how much farther will it travel than if it flew along the shortest path? ([See Fig. 7.]) (A path always at the same angle from the north is called a loxodrome, and a shortest path on a globe is called a great circle.)
How should an engine be designed so that it will have the least vibration when it moves fast?
In physical problems like these, the answer is not a single number but a formula. What we want to do in any one of these problems is find a formula so that any one of the quantities may be calculated, given the behavior of the others. For example, here is a familiar problem in which the answer is a formula and not a number:
Fig. 7. Paths of a flight.
Fig. 8. Room formulas.
How are the floor area of a room, its length, and its width related to each other? ([See Fig. 8.])
The answer is told in any one of three equations:
(floor area) equals (length) times (width)
(length) equals (floor area) divided by (width)
(width) equals (floor area) divided by (length)
The first equation shows that the floor area depends on the length of the room and also on the width of the room. So we say floor area is a function of length and width. This particular function happens to be product, the result of multiplication. In other words, floor area is equal to the product of length and width.
Now there is another kind of function called a differential function or derivative. A differential function or derivative is an instantaneous rate of change. An instantaneous rate of change is the result of two steps: (1) finding a rate of change over an interval and then (2) letting the interval become smaller and smaller indefinitely. For example, suppose that we have the problem:
How are speed, distance, and time related to each other?
One of the answers is:
(speed) equals the instantaneous rate of change of (distance) with respect to (time)
Or we can say, and it is just the same thing in other words:
(speed) equals the derivative of (distance) with respect to (time)
Now we can tell what a differential equation is. It is simply an equation in which a derivative occurs, such as the last example. Perhaps the commonest kind of equation in physical problems is the differential equation.