The three Rectilinear Equations of motion:

So far, you learned about displacement, motion with constant or variable velocity, acceleration. Here we are considered with motion with constant acceleration (i.e. velocity changes linearly with time).

So now, when it comes to acceleration, the motion can be complex and difficult to analyse. There are a variety of quantities associated with the motion of objects - displacement (and distance), velocity (and speed), acceleration, and time. Knowledge of each of these quantities provides descriptive information about an object's motion. For example, if a car is known to move with a constant velocity of 22.0 m/s, for 12.0 seconds for a displacement of 264 meters, then the motion of the car is fully described. We’re going now to

Let us start by trying to figure out a relation between velocities at different instances.

We know from acceleration definition that it means change of velocity over time; we know that we are now working on constant acceleration; so the instantaneous acceleration at any time will be equal to the average acceleration; thus

Where we suppose that initial time will be zero so final time will be t, by multiplying both sides with t we get:

Now, we have a relation giving us the final velocity at any instant t, if we have the initial velocity, and of course the value of the constant acceleration, (i.e. Vf here is defined in terms of time; keep it in mind).

Previously, in constant velocity motion, if you want to find the displacement after any time, you can easily get it though (V x t). The velocity was constant so that the displacement increased in a linear growth with time (i.e. for each portion of time the displacement increases with a constant value).

Consider the following example:

You notice here that after each constant portion of time (here: 1 sec), the displacement increases with a constant value (here: 10 m).

Try to consider then what happens in case of accelerated motion (i.e. velocity changes) you will find that the amount of increase in displacement is changing along different portions of time.

In this case if we want to get the displacement (x), trying to use the above relation of (x = v t) we get a question: Which velocity?

To get the relation of displacement, you have two ways:

First:

From the relation we know (x = v t); we are facing a question of which velocity to use; as velocity varies with time. We notice that for any variable quantity we can get an average value that affects the same change as the variable one.

Here, we have a linear growing velocity (acceleration constant); thus, the average can be expressed as the arithmetic mean of the initial and final velocities:

Substitute in (x = v t) by this average velocity, you get:

However, we aim to get x in terms of time and initial velocity, we got from the first equation a relation between final and initial velocities, substitute it in the prev. equation you get:

Second:

You sure noticed that we were mentioning many times a “variable” velocity, we needed to use the relation (x = v t) but the velocity is “variable”!

What to do then? Here comes the role of calculus...

A main application of Integration in calculus is to “multiply” variable quantities (i.e. if v is constant we say x = v t, but when v is variable over or along time, then it becomes ).

Taking into consideration that:

Then:

Evaluating the integral gives:

Notice that: for the equation: if we plotted it graphically, it gives us this graph:

From calculus: we know that the Integration of v over time gives us the area under the st. line; thus we can consider the displacement (x) at any instant t = Area under the line…

So Area = (Area of green rectangle) + (Area of grey triangle)

Since: Area of green rectangle = Length x width =

Since: Area of grey Triangle = ½ x base x height = ½ x x =

Total area = Displacement =

Keeping up: Until now, we derived two Equations for rectilinear motion (i.e. motion in straight line with constant acceleration):

First:

Second:

However, these two equations give us velocity or displacement in terms of time. How about getting velocity in terms of another parameter? Displacement for example (i.e. getting final velocity after travelling 40m for example).

We found that by some algebraic formulations, we could find such equation.

First, we start by:

Squaring both sides, we get:

By taking “2a” as a common factor from the second and third term, we get:

Noticed something remarkable??! Yes, the value in the rectangle is equal to the displacement (Derived in second equation). Thus, we get:

This is the Third equation of rectilinear motion, where the final velocity described in terms of distance and initial velocity.

References:

1- Stewart Calculus Early Transcendental, 7th Edition.2- Serway & Jewett Physics, 9th Edition.