Understand concepts of rectilinear motion in a straight line.
Rectilinear Motion of Particles in 1 Dimension
Motion, in general relates to the study of positing and rate of change of position of a particle in space. In this article we will learn about motion and the equations governing motion in a straight line also called as rectilinear motion.
Introduction to Motion
An object is defined to be in motion with respect to a reference point if there is a change in its position with respect to time. The reference point is generally considered to be the origin. Motion refers to the position of the object in 3-dimensional space, as well as the rotation about its axis. It is characterized by vector quantities like position, displacement, velocity and also acceleration. If the object’s motion is restricted to one dimension, or a line, it is considered to be in rectilinear motion. This article first covers the various types of motion, then the theory, and finally the math that governs bodies in rectilinear motion.
Our Nature consists in Motion; complete rest is death.
– Blaise Pascal
Types of Motion
Broadly speaking, all motion can be classified into a combination of translatory and rotatory motion, i.e. the change in the position of the object in space and the rotation about its axis, however, motion can be further categorized into simpler groups for a better understanding. These different types of motion have their own unique math that is simplified from the basic laws of motion.
Translational Motion
Translational motion is essentially, a type of motion where an object moves along a straight line or curved path without rotating. In translational motion, every point of the object moves the same distance in the same direction during each time interval.
Rectilinear Motion
Rectilinear motion is indeed the motion of an object moving in a straight line. It is a type of translational motion. It is also referred to as motion in one dimension. Since the motion is restricted to one dimension, its concepts can be understood with minimal knowledge of vector mathematics.
Examples:
- A free falling body.
- The movement of a bead on a tight string.
Curvilinear Motion
Curvilinear motion is specifically, the motion of an object along a curved path. Unlike rectilinear motion, where movement is in a straight line, curvilinear motion occurs when an object’s trajectory bends in any direction. Furthermore, Curvilinear motion occurs in two or three dimensions.
Examples:
- Trajectories of asteroids.
- Projectile motion.
- The movement of an ant.
Rotatory Motion
Rotatory motion (or rotational motion) is the motion of an object about a fixed axis. Moreover, in this type of motion, different points on the object follow circular paths around the axis of rotation. Unlike translational motion, in rotatory motion, different parts of the object move different distances in the same time interval.
Examples:
- The spinning of a ceiling fan.
- The rotation of the Earth about its axis.
- A rolling wheel.
Periodic or Oscillatory Motion
Periodic motion is a type of motion that repeats itself at regular time intervals. Oscillatory motion is when an object moves back and forth around a fixed point in a regular pattern. Furthermore, it is a type of periodic motion.
Examples:
- The simple harmonic motion of a pendulum.
- Vibrations of a tuning fork.
- The oscillation of a spring.
Circular Motion
Circular motion is a type of motion in which an object moves along a circular path. It can be uniform (constant speed) or non-uniform (changing speed). If an object experiences a force directed toward the center, it follows a circular trajectory due to centripetal acceleration.
Uniform Circular Motion is a type of curvilinear motion as well as periodic motion.
Examples:
- A satellite orbiting the Earth.
- A car taking a roundabout.
- A spinning top.
Random Motion
Random motion is an unpredictable type of motion in which an object moves in an irregular and erratic manner, changing direction and speed frequently. Microscopic particles and natural phenomena often display this type of motion.
Examples:
- The motion of dust particles in the air.
- Brownian motion.
Basics of Rectilinear Motion
Position
Definition
Position is the location of an object in a given frame of reference relative to a fixed point (usually called the origin). It is typically described using coordinates in one, two, or three dimensions.
Mathematical Intuition
In a 1-D system, position is represented as:
![\[x(t)\]](https://entechonline.com/wp-content/ql-cache/quicklatex.com-8d7ec9643b767c61116ade460a963efe_l3.png "Rendered by QuickLaTeX.com")
where 
is the position as a function of time and 
is the time of consideration.
Position in 1-D is graphically represented on a number line, where the magnitude is scaled and the right side is considered positive while the left side negative.
Consequently, in higher dimensions, the position vector is represented by both magnitude and direction.
![\[r(t)=x \mathbf{\hat{i}} +y\mathbf{\hat{j}}+ z\mathbf{\hat{k}}\]](https://entechonline.com/wp-content/ql-cache/quicklatex.com-28079e20a5284975c190ccad466337c0_l3.png "Rendered by QuickLaTeX.com")
where 
are unit vectors along 
axes.
Distance
Definition
Distance is the total length of the path traveled by an object, regardless of direction. It is a scalar quantity, meaning it has magnitude but no direction.
Mathematical Intuition
For a straight-line motion in one dimension:
![\[d=|x_2 - x_1|= \sqrt{(x_{2} - x_{1})^{2}}\]](https://entechonline.com/wp-content/ql-cache/quicklatex.com-d51711d054f5a122595525f7bcb1ec1f_l3.png "Rendered by QuickLaTeX.com")
where 
and 
are the initial and final positions.
Since distance only depends on the path length, it is always positive and does not decrease with direction changes.
Displacement
Definition
Displacement is the shortest path or straight-line distance from an object’s initial position to its final position. It is a vector quantity, meaning it has both magnitude and direction. Distance is always positive while displacement can be positive or negative.
Mathematical Intuition
In one spatial dimension, displacement is:
![\[\Delta x = x_2 - x_1\]](https://entechonline.com/wp-content/ql-cache/quicklatex.com-bad76e270ee3e4345283cf0c07c061f6_l3.png "Rendered by QuickLaTeX.com")
where is the initial position and is the final position.
Time Dependence of Rectilinear Motion
Speed and Velocity
Definition
- Speed is the rate at which an object covers distance. It is a scalar quantity. Therefore, it only has magnitude and no direction.
- Velocity is the rate of change of displacement. Since velocity is a vector quantity, it has both magnitude and direction.
Mathematical Intuition
For an object moving a certain distance 
in time , the speed is:
![\[Speed = \frac{\text{Distance traveled}}{\text{Time taken}}=\frac{|d|}{t}\]](https://entechonline.com/wp-content/ql-cache/quicklatex.com-beb6e5a6a62e3bfd375eaee7a78b4532_l3.png "Rendered by QuickLaTeX.com")
For an object whose displacement is 
in time 
, the velocity is:
![\[Velocity = \frac{\text{Displacement}}{\text{Time taken}}=\frac{d}{t}\]](https://entechonline.com/wp-content/ql-cache/quicklatex.com-c26c50d5b3a3cd69c52ddf0643d5aa19_l3.png "Rendered by QuickLaTeX.com")
Since velocity depends on displacement, it considers direction, whereas speed does not.
Average Speed and Average Velocity
The total distance traveled divided by the total time taken:
![\[Speed_{Average} = \frac{\text{Total Distance }}{\text{Total Time}}=\frac{\Sigma |d|}{T}\]](https://entechonline.com/wp-content/ql-cache/quicklatex.com-2b129622a13f09f11716b9c193e1e463_l3.png "Rendered by QuickLaTeX.com")
Since distance is always positive value.
The total displacement divided by the total time taken:
![\[Velocity_{Average} = \frac{\text{Net Displacement }}{\text{Total Time}}=\frac{\Sigma d}{T}\]](https://entechonline.com/wp-content/ql-cache/quicklatex.com-3faef7cfc92144dd1d3d4552da6a558c_l3.png "Rendered by QuickLaTeX.com")
Since displacement can be positive, negative, or zero, average velocity can also be positive, negative, or zero.
Key Difference: If an object returns to its starting point, displacement is zero, making average velocity zero, but the average speed is nonzero because distance traveled is not zero.
Instantaneous Velocity
Instantaneous velocity is the velocity of an object at a specific moment in time. Hence, it is the rate of change of displacement with respect to an infinitesimally small time interval.
Instantaneous velocity is obtained by taking the limit of the average velocity as the time interval approaches zero:
![\[v=\frac{dx}{dt}\]](https://entechonline.com/wp-content/ql-cache/quicklatex.com-f495ba69630478a8510a8479c2c8722d_l3.png "Rendered by QuickLaTeX.com")
This is the first derivative of position with respect to time meaning it gives the exact velocity at any given instant.
If velocity changes over time, its instantaneous value is different at different moments, and we use differentiation to find it.
Hence velocity is the time derivative of displacement and can be calculated at any instant by measuring the slope of the curve on a displacement versus time graph.
Accelerated Motion along a Straight Line
Acceleration
Definition
Acceleration is the rate of change of velocity with respect to time. It is a vector quantity, meaning it has both magnitude and direction. An object accelerates when its velocity changes, either in magnitude (speed) or direction.
Uniform rectilinear motion is when a particle moves in a straight line with a constant velocity. Consequently, the linear motion of a particle as stated above is described to be under zero acceleration.
Mathematical Intuition
The average acceleration over a time interval is given by:
![\[a_{avg}=\frac{\Delta v}{\Delta t}=\frac{v_2-v_1}{t_2-t_1}\]](https://entechonline.com/wp-content/ql-cache/quicklatex.com-cdcf7d7442dc40e449bf5c618324d984_l3.png "Rendered by QuickLaTeX.com")
where 
and 
are initial and final velocities, respectively.
The instantaneous acceleration is the derivative of velocity with respect to time:
![\[a=\frac{dv}{dt}\]](https://entechonline.com/wp-content/ql-cache/quicklatex.com-e5e7d6f6c67431741c3e1bfc5da0278a_l3.png "Rendered by QuickLaTeX.com")
Since velocity is the derivative of displacement, acceleration is the second derivative of displacement:
![\[a=\frac{d^{2}x}{dt^{2}}\]](https://entechonline.com/wp-content/ql-cache/quicklatex.com-a3490d7715be502984e00a79d7136e59_l3.png "Rendered by QuickLaTeX.com")
where is displacement.
Hence acceleration is the time derivative of velocity and is equivalent to measuring the slope of the curve on a velocity versus time graph .
- Positive acceleration increases velocity in the direction of motion.
- Negative acceleration (deceleration) decreases velocity.
Kinematic Equations
For uniformly accelerated rectilinear motion (motion with constant acceleration) , the following kinematic equations describe relationships between displacement, velocity, acceleration, and time:
Velocity-Time Relation:
![\[v=u+at\]](https://entechonline.com/wp-content/ql-cache/quicklatex.com-6693aa3656d9280bf409b96e1d078203_l3.png "Rendered by QuickLaTeX.com")
where 
is initial velocity, 
is final velocity, 
is acceleration, and is time.
Displacement-Time Relation:
![\[S=ut+\frac{1}{2}at^{2}\]](https://entechonline.com/wp-content/ql-cache/quicklatex.com-83aba77ca37cc739d52902ed25750b05_l3.png "Rendered by QuickLaTeX.com")
where 
is displacement.
Velocity-Displacement Relation:
![\[v^{2}=u^{2}+2aS\]](https://entechonline.com/wp-content/ql-cache/quicklatex.com-43606f3aa2e1831791872fa0bab350d9_l3.png "Rendered by QuickLaTeX.com")
Note: The above relations only hold true when acceleration is constant (not necessarily uniform motion) and fail to describe any motion with non-uniform acceleration.
I can calculate the motion of heavenly bodies but not the madness of people.
-Sir Isaac Newton
Relative Displacement
Definition
Relative displacement is the displacement of one body with respect to another. It tells us how the position of one object changes relative to another moving object.
Mathematical Intuition
If two objects A and B have position vectors 
and 
, the displacement of A relative to B is:
![\[x_{AB}=x_A-x_B\]](https://entechonline.com/wp-content/ql-cache/quicklatex.com-adfeee1c428f202fbcfe61e2bd993f15_l3.png "Rendered by QuickLaTeX.com")
This vector gives both the magnitude and direction of how A is positioned with respect to B.
If both objects are moving, their relative displacement can change over time.
Relative Velocity
Definition
Relative velocity is the velocity of one object as observed from another moving object. It shows how fast and in what direction one object is moving relative to another.
Mathematical Intuition
If two objects A and B have velocities 
and 
, then the velocity of A relative to B is:
![\[v_{AB}=v_A-v_B\]](https://entechonline.com/wp-content/ql-cache/quicklatex.com-574b95d6b0428c10ed54df4528de7667_l3.png "Rendered by QuickLaTeX.com")
Similarly, the velocity of B relative to A is:
![\[v_{BA}=v_B-v_A\]](https://entechonline.com/wp-content/ql-cache/quicklatex.com-9299f4ede641223b5cf5a395d1d6c14b_l3.png "Rendered by QuickLaTeX.com")
Since 
, the relative velocities are equal in magnitude but opposite in direction.
Speed is relative. You have to live it. You can’t just jump into it. You have to live it all the time.
– Mario Andretti
Conclusion
Rectilinear motion is the foundation of kinematics, describing motion in a straight line under constant or varying acceleration. Hence, by applying kinematic equations, we can analyze velocity, acceleration, and displacement for different scenarios, from free-fall motion to uniformly accelerated motion. Therefore, understanding rectilinear motion is essential for solving real-world problems in physics, engineering, and everyday life.
Frequently Asked Questions
Q. What is rectilinear motion of particles?
Rectilinear motion refers to the motion of a particle along a straight line. Parameters such as velocity, displacement, and acceleration describe this type of motion. In rectilinear motion, the path taken by the particle does not change direction; it moves either forward or backward along the same line. This makes it distinct from other types of motion, such as rotational motion or oscillatory motion.
Q. What are the key characteristics of uniform rectilinear motion?
Uniform rectilinear motion occurs when a particle moves along a straight line with a constant speed. In this case, the velocity of the particle remains unchanged over time, and the displacement is directly proportional to the time elapsed. Hence, rate of change of displacement with respect to time is constant, leading to a linear relationship between displacement and time.
Q. What is accelerated rectilinear motion?
Accelerated rectilinear motion is when a particle moves along a straight line with a changing velocity. This type of motion can be further categorized into uniformly accelerated rectilinear motion, where the acceleration is constant, and non-uniform acceleration, where the acceleration varies over time. In uniformly accelerated rectilinear motion, the equations of motion can be easily derived, allowing for straightforward calculations of displacement, velocity, and time.
Q. What are the formulas associated with rectilinear motion?
The primary formulae used in rectilinear motion include:
References
- Elsharkawy, K., & Ammar, M. K. (2023). Chapter 1—Rectilinear motion of a particle. In Kinematics of a particle. https://www.researchgate.net/publication/371938315_Chapter_1-Rectilinear_Motion_of_a_Particle_CHAPTER_1_KINEMATICS_OF_A_PARTICLE
- Routh, E. J. (2014). Rectilinear motion. In A treatise on dynamics of a particle (pp. 55–76). Cambridge University Press. https://doi.org/10.1017/CBO9781139237277.004
- Whittaker, E. T. (1988). Rectilinear motion. In A treatise on the analytical dynamics of particles and rigid bodies (4th ed., pp. 1–25). Cambridge University Press. https://doi.org/10.1017/CBO9780511608797.003
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