# How does acceleration relate to distance?

May 19, 2015

Acceleration is the second derivative of distance with respect to time. If the motion is along one dimension ($x$) we can write:

$a = \frac{{d}^{2} x}{\mathrm{dt}} ^ 2$

The first derivative is velocity. That determines how fast the distance is changing. If someone is moving away from you at 1 meter per second, the distance away from you changes by one meter every second.

$v = \frac{\mathrm{dx}}{\mathrm{dt}}$

If the velocity is constant, then the second derivative will be zero. Zero acceleration means that the velocity is not changing. But if the other person is either speeding up or slowing down, the second derivative will be non-zero and we say that they are accelerating. If the acceleration is decreasing the magnitude of the velocity, we might say that they are decelerating.

A ball thrown into the air will experience a constant acceleration in the downward direction. The velocity is initially large in the upward direction (usually positive). At the top of its flight the velocity becomes zero. And then the ball begins to fall and the velocity becomes larger (or more negative) until it is caught or hits the ground. Though the distance and the velocity are constantly changing, the acceleration of an object in freefall is always constant.

May 19, 2015

This is a complicated question! I'll try (and probably confuse you even more!!

Consider an object that moves, for example, along the $x$ axis. The object continuously change position and in doing so covers a certain distance. You can have now a lot of different objects covering the same distance (a car, a man, a dog, a snail a rocket...etc.).

Each object takes a certain TIME do cover the same distance. If you want to discriminate between the snail and the rocket you say that the rocket takes less time to cover the distance, or, the rocket is faster than the snail. This is velocity i.e. the RATE of CHANGE of position with time!

Now, forget the snail, and consider two cars. Both of them are fast and can reach a certain velocity, say, $100 \frac{k m}{h}$ but the first car is my old Fiat 500 and the other is a F1 Ferrari...ok, they both can reach $100 \frac{k m}{h}$...(ok, let's assume my car can reach it :-(!!!) so what is the difference between the two? It is acceleration !

The Ferrari reaches $100 \frac{k m}{h}$ in few seconds...my car in...few hours...maybe! So acceleration tells us how velocity changes with time i.e. it is the RATE of CHANGE of velocity with time!

Now the difficult bit:

In mathematics, and Calculus in particular, we invented a way to evaluate these RATES of CHANGE, called DERIVATIVE .

I do not know if you already studied this but between us let's say that derivative is a strange operation that tells us how a "thing" changes.

If you have displacement it can be enclosed into a function (a little mathematical box that when you input time gives you position) so if you want velocity you use the derivative of this displacement in time....if you want acceleration you do it again!!! You do the (second) derivative of displacement with time!!!

So acceleration is connected, through this operation of finding rates of changes, to displacement and tells us how fast you are fast in covering a certain fixed distance!!!!!!!

Visually:

Hope it helps!