Motion In A Straight Line - the Geometry of Motion

Hello everyone!

Today we are going to discuss and understand every aspect of “motion in a straight line”. It’s an important part of Kinematics – “the geometry of motion”.

Before taking a deeper dive into this concept, let’s see WHY do we need to study this and WHERE will we use this concept of Kinematics – “motion in a straight line”.

WHY And WHERE?

“Why”, because it lets us know how movement of various things in worlds of one dimension works, and by understanding these we can make use of it into different kinds of innovative inventions through engineering.

Everyone has traveled through trains at least once in their life, if not – then your first journey awaits you.

When we arrive at a railway station before time, we always keep our eye on the display panel that describes the arrival time of trains. So, how do station masters update the display panels with accurate timings of train arrival? Could it be that station masters have foresight?

Hahaha, that’s not it. They just get the train’s current position or distance and speed by communicating with the train driver, then input in a time calculator software (which uses formulas described in kinematics – “motion in a straight line.” Nowadays, these things are completely automated and are done by automated softwares and bots.

This is just one example where the concept of kinematics and motion in straight line is being used in real time, but there are many such examples in different fields like satellite and rocket trajectory analysis, interstellar debris and asteroids, free fall, car driving on a straight road, etcetera. Not everything follows a straight line path, for these we’ll also learn “motion in a plane.”

We need to study this because it’s a basic part of classical mechanics and also to understand the physical world around us.

Now that we get to know how important this concept is, let’s start studying in depth with proper derivations of all the equations of motion in one dimension.

It’s not too late to know that motion in a straight line is the same as one dimensional motion.

As the name suggests, it’s the motion where the object only travels in a straight line, either forward or backward, no left or right.

Convention In One Dimensional Motion

In a coordinate system, we generally take the x axis as the line of motion or displacement from origin (starting point, position of object at time – t = 0) and y axis for time.

When a body moves in the positive direction of x axis, we say it has positive velocity and acceleration, similarly if a goes in the negative direction of x axis, we call it by negative velocity and acceleration.

Co-ordinate System Illustration.

Don’t be confused with these positive and negative sign conventions, it’s done because after using these in equations, we get to know the direction of resultant magnitude of velocity and acceleration.

When dx/dt is positive, the direction of velocity will be along positive x axis and if dx/dt is negative, the direction will be along the negative x axis, just the opposite. Similarly, if dv/dt is positive, the acceleration is along positive x axis and if dv/dt is negative, the acceleration is along the negative x axis.

Velocity and acceleration formula in derivative form.

In one dimensional motion, if the velocity and acceleration share the same sign, then the speed of the body increases, but if their sign is different, the speed of the body decreases and the direction can be determined through sign convention.

Equations Of Motion

These equations of motion were proposed by Sir Isaac Newton. And these only apply when an object moves with constant acceleration in a straight line.

Equations of motion.

In the derivation, we assume that the acceleration of the moving body remains constant throughout the time – t.

First Equation Of Motion

Let’s assume that at time – t1 = 0, the velocity of a body is u (initial velocity) and at time – t2 = t, the velocity is v (final velocity) and this change is done over a constant acceleration – a.

figure shows how we got our first equation of motion.

The derivation includes some basic integrations, which is very easy to understand.

Now, with the help of the first equation of motion, we’ll derive our second equation of motion.

Second Equation Of Motion

In this equation, we will focus on a relation to find distance, when the other variables are known. Of course, the assumption made in the first equation of motion holds true here as well, so we’re carrying it forward.

We are starting by breaking down the first equation of motion, where velocity on the left hand side is being broken down into distance (x) per unit of time (t).

figure shows the step by step derivation of second equation of motion.

The above figure shows the derivation of the second equation of motion. It’s pretty easy and no complex calculus operations are used.

By using this equation we can determine the distance of any moving body with a constant acceleration (necessary condition for equations of motion).

Third Equation Of Motion

The third equation of motion tells us the relation between all the four variables – initial velocity, final velocity, distance travelled and the constant acceleration throughout, and more importantly we don’t have a variable of time in our equation, so we don’t need time in order to calculate these values.

This equation can also be derived by using the first and second equation of motion.

First we need to take the square to both sides of the first equation of motion, the rest is manipulating the equation to eliminate the time variable, through proofs and verifications.

figure shows the step by step derivation of the third equation of motion.

The above figure shows the stepwise derivation of the third equation of motion, here we have replaced the expression under the brackets with distance – x, using the second equation of motion.

Please note that these equations are based on the coordinate system and in general don’t tell the absolute distance, but the difference in coordinates, we can get the distance by finding the distance between the coordinate points. That’s why the values of u, v, a, or x can be negative or positive as per the direction in the coordinate system. But it doesn’t apply on the time – t, because we know it’s not something that’s measured on this coordinate system.

We can manipulate the positive and negative directions in the coordinate system as per our choice, but make sure to use that same coordinate system throughout the calculation.

Freely Falling Bodies

Freely falling bodies is one such condition where we can apply what we have learned till now. Freely falling bodies also follow the motion in a straight line, since they are falling freely and nothing is coming in their path to deviate them from their trajectory.

What happens in this case is that the body falls solely under the effect of the gravitational force of the earth. So you can say that the value of acceleration in the equations can be replaced by gravity – g, which is about 9.8 ms^-2 on the surface of earth.

The diagram below shows the equations of motion, customized for the freely falling bodies.

Equations of motion for freely falling bodies.

So, with that we have covered the science behind the motion in a straight line. Next time we will see about the science behind the motion in a plane or two dimensions.