Calculus revolutionised maths and physics in the 17th century, allowing Isaac Newton to form his equations of motion. This article will lay out the physical meaning of its primary components, differentiation and integration, and show their power by, from first principles, deriving Newton’s equations of motion.
Differentiation finds the slope of a curve. In others words, by how much y changes when x is increased by 1. For any function:The slope (or derivative) is given by:For f(x) = x, as plotted below, the gradient f'(x) = 1, meaning that y increases the same as x, which makes sense.Differentiation is therefore the rate of change. The rate of change of position is velocity, and the rate of change of velocity is acceleration.
Integration, on the other hand, finds the curve from the slope. It is the opposite of differentiation. For any function:
The curve (or integral) is given by:So for f'(x) = 1, the integral f(x) = x + c. The c is an uncertainty constant. This arises because knowing the slope isn’t enough to know where the curve starts and ends, just as knowing the velocity of a car gives no information about its position. Yet the integral of velocity is position, and the integral of acceleration is velocity. This last one is where we begin.
Define acceleration as a. From first principles the velocity v will be the integral of a with respect to t. Performing this integral returns:In other words, the a gains a t term (as expected) and there is an uncertainty constant u. Now what is u? Well, if t = 0, we can see that v = u, and so u is the initial velocity, before time commenced. This is one of Newton’s equations of motion, but he has another.
If we integrate again, we will arrive at position:
Where, again, x₀ is the uncertainty constant and is equal to x at t = 0. Thus it is the initial position. By redefining x – x₀, (i.e. distance between final and starting positions) as the displacement s, the second of Newton’s equations of motion is derived.