Motion in One Dimension

There are several ways to describe how an object is moving. Speed, velocity, acceleration and displacement are a few. Three of the aforementioned are vectors while the other is a scalar. Displacement is the change in distance. It has a magnitude and a direction. For example if something is 4 meters from me and then it comes to me it will have traveled -4 meters assuming that the object was on the positive x direction with respect to me.

The average velocity is the displacement over the time. So it too is a vector. In the previous example, if the object had taken 2 seconds to travel the distance. It would have gone at an average velocity of -2 m/s with respect to me. Speed is the actual distance traveled over time. In this case, the distance traveled was four meters as distance is always positive, though displacement is not. Thus, the speed is equal to two meters per second. If a plane were to fly around the world and come back to its starting position, it would have a zero average velocity as its displacement is 0. It would however have a non-zero average speed as it circumnavigated the globe and traveled a large distance.

The average acceleration is equal to the change in velocity over time. If an object was thrown with an initial velocity of 10 m/s and landed with a velocity of 0 m/s ten seconds later, it would have an average acceleration of 10 m/s^2.
In more mathematical terms. the following describe previously mentioned terms.

Through some derivations, it can be shown that