Rectilinear motion is the motion of an object along a straight line, in which its position at any instant is given by a single coordinate $x$ along that line. The relationship between this coordinate and time, written as $x = f(t)$, is called the equation of motion, and it is usually found by fitting a function to position-time data recorded experimentally at equal time intervals.
We all know the basic idea of motion, and it plays a central role in building our scientific understanding of the world. Rectilinear motion is the motion of a particle along a straight line. It is the simplest type of motion often studied in physics, and it forms one of the foundational ideas within the broader topic of motion, rest, and frames of reference.
The frame of reference is an important concept when we study any kind of motion. It lets us define the position and velocity of an object relative to a particular observer or coordinate system.
In this article, we look at the idea of rectilinear motion and the method used to find the equation of motion of a moving object.
What is Rectilinear Motion?
Rectilinear motion occurs when an object moves along a straight line.
- For simplicity, we align a coordinate axis (say, the x-axis) along the object’s path.
- At any given moment, the moving object has a definite coordinate relative to this frame of reference.
- The object’s coordinate is a function of time, written as $x = f(t)$.
So, the function $f(t)$ that gives the position of the object at every instant is called the equation of motion.
How We Determine Motion Equations Experimentally
To find motion equations experimentally, we mark an object’s position on the coordinate axis at fixed time intervals.
Consider a moving car as an example. We place markers along the road and record the car’s position relative to these markers at equal time intervals.
Let us assume that the table below shows the result of this experiment. We will be using this data to understand the car’s motion at different instants of time.
| Time $t$ (s) | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
|---|---|---|---|---|---|---|---|---|
| Position $x$ (m) | 1 | 1.25 | 1.50 | 1.75 | 2.00 | 2.25 | 2.50 | 2.75 |
Note that this experiment does not tell us the object’s position at intermediate moments — for example, between $t=1\text{ s}$ and $t=2\text{ s}$.
To overcome this, we look for a function $x = f(t)$ that fits the recorded data points, where $t$ is time and $x$ is the object’s position at that time.
From the table:
- When $t = 1\text{ s}, x = 1\text{ m}$
- When $t = 2\text{ s}, x = 1.25\text{ m}$
- When $t = 3\text{ s}, x = 1.50\text{ m}$, and so on
The position increases by $0.25\text{ m}$ for every $1\text{ s}$ that passes. This is a constant rate of change, so $x$ is a linear function of $t$.
Thus, using the slope $0.25\text{ m/s}$ and the point $t=1\text{ s}, x=1\text{ m}$, the equation of motion is:
$$x = f(t) = 0.75 + 0.25t$$
As long as the object’s motion stays consistent, this equation can be used to find the position of the car at any moment between the first and eighth second — the interval over which the experiment was conducted.
Importance of Time Intervals and Volume of Data
The accuracy of a motion equation depends heavily on how much data we collect.
- The more markers we use, and the shorter the time interval between them, the more accurately we can determine the motion equation.
- Sufficient data is essential to keep the derived equation close to the object’s true motion.
Going back to the car example, the equation of motion we derived is valid only for the time interval over which the experiment was conducted. In other words, it holds only as long as the conditions of motion — such as the car’s speed — stay the same throughout that interval.
Extrapolation means using a motion equation to predict the object’s position beyond the interval that was actually investigated. This is risky if the character of the motion changes, since the equation may no longer describe the object’s behaviour correctly outside the experimental time frame.
To extend the validity of a motion equation beyond the original interval, we need supplementary investigations. These additional experiments confirm that the character of motion has not changed, before the equation is applied further.
Limitations of Motion Equations
A motion equation determined from experiment is valid only for the time interval over which the investigation was carried out.
IMPORTANT NOTE:
- An equation of motion is only reliable within the investigated interval.
- Extrapolating it beyond that interval is only valid after supplementary investigation confirms the motion’s character has not changed.
- If the character of motion changes (for example, the car’s speed changes), the old equation no longer describes the motion accurately beyond that point.
Consider the car example again. We assumed the car’s speed stayed constant throughout the experiment, which let us derive one clean equation of motion.
Now suppose that, after the eighth second — beyond the interval we investigated — the driver accelerates or decelerates the car. This changes the character of the car’s motion, so the equation derived above is no longer valid past that point.
Using the original equation to predict the car’s position after this change would give an inaccurate answer. To find a new equation valid for this later interval, we would need to conduct another experiment and collect fresh data under the new conditions.
Conclusion
Rectilinear motion is a fundamental concept in physics. Understanding it requires a grasp of motion equations and the frame of reference. By running experiments and collecting accurate data, we can determine equations that correctly describe how an object moves — and know exactly the limits within which those equations hold.
Frequently Asked Questions
What is rectilinear motion?
Rectilinear motion is the motion of an object along a straight line. It is the simplest type of motion studied in physics, since only one coordinate axis is needed to describe the object’s position at any instant.
What is an equation of motion?
An equation of motion is a function $x = f(t)$ that gives the position of an object at any instant of time. It is usually found by fitting a function to experimentally recorded position-time data.
How do you find the equation of motion from experimental data?
Record the object’s position at equal time intervals, then look for a function that fits all the recorded data points. If the position changes by a constant amount for each equal time step, the motion is described by a linear equation.
Why can’t a motion equation always be extrapolated?
A motion equation is only valid for the time interval over which it was derived. If the character of motion changes afterward — for instance, if the object speeds up or slows down — the same equation no longer describes its position accurately.
What is the difference between interpolation and extrapolation in motion equations?
Interpolation estimates the object’s position at an intermediate time within the investigated interval, where the equation is known to hold. Extrapolation estimates its position beyond that interval, which is reliable only if supplementary investigation confirms the motion’s character has not changed.
References and Further Reading
- B. M. Yavorsky, A. A. Pinsky — Fundamentals of Physics, Vol. 1, MIR Publishers (1975)
- Rest and Motion in Physics
- Isotropy and Uniformity in Space
- Uniformity of Time in Physics
