Rectilinear Motion of Particles

What is Rectilinear Motion?

If the position of an object changes with respect to time and its surroundings, the body is said to be in motion. Mathematically, motion can be described with displacement, velocity and acceleration in a particular frame of reference. The motion of a particle can be classified on the basis of its trajectory, the simplest being motion along a straight line namely rectilinear motion. The displacement, velocity and acceleration vectors are restricted to one dimension. Rectilinear motion has three types: uniform motion (zero acceleration), uniformly accelerated motion (non-zero constant acceleration) and motion with non-uniform acceleration. Examples of rectilinear motion are free-fall under gravity and simple harmonic motion of a mass attached to a spring.

Rectilinear Motion Definition

If a particle is restricted to move along a straight line, its motion is called rectilinear (or linear) motion. Such a motion can be described using one coordinate only. Displacement of the particle and its derivatives i.e. velocity and acceleration all are one-dimensional vectors. Free-fall under the Earth’s gravitational field, a car moving along a straight path can be approximated as rectilinear motions. 

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Mathematical Form of the Motion

To qualitatively study a rectilinear motion, a one-dimensional reference frame consisting an axis (X-axis) and an origin at O (x = 0) is considered.

Position, distance and displacement: The position of a particle is a vector quantity which points from the origin to the particle. Its magnitude is given by the distance between them. When the particle is set into motion, it follows a path so that the position changes with time. Displacement is the vector difference of the position after an interval of time and it points from the initial position to the final position. Distance is the total path traversed along the trajectory whereas displacement is the shortest path. If the position of the particle changes from xi  to xf in time  \[\Delta t\], the displacement is given by, 

\[x = {x_f} - {x_i}\]

Speed and Velocity: Rate of change of distance is called speed and the time rate of change of displacement is called velocity. Speed is a scalar but velocity is a vector having direction same as displacement. Instantaneous velocity at a time t is given by,

\[v = \frac{d}{{dt}}x\]

Acceleration: If velocity changes with time, the time rate of change is defined as acceleration. It is also a vector,

\[a = \frac{d}{{dt}}v = \frac{{{d^2}}}{{d{t^2}}}x\]

Since all the vectors are restricted to one dimension, it is enough to consider the magnitudes only. 

Graphical Representation

If position is plotted as a function of time, the graph shows the trajectory of the particle. Velocity at any instant is given by the slope of this graph since velocity is the time derivative of position. Acceleration is the time derivative of velocity so it is given by the slope of velocity versus time graph.

Rectilinear Motion Formulas Derivation

Considering different values of acceleration, rectilinear motion can be categorized into three types which are: uniform rectilinear motion, uniformly accelerated motion and motion with non-uniform acceleration.

  • Uniform Rectilinear Motion Definition: It describes a motion along a straight line with zero acceleration. The velocity of the particle does not change with time such that,

\[\frac{{dv}}{{dt}} = 0\]

\[v = {v_o}\]

v₀ is the constant velocity. From the above equation,

\[\frac{{dx}}{{dt}} = {v_o}\]

\[\int_{{x_o}}^x {dx}  = {v_o}\int_0^t {dt} \]

\[x\left( t \right){x_o} + {v_o}t\]

Here, x₀  is the initial position of the particle i.e.  \[x\left( {t = 0} \right) = {x_o}\] Clearly, the trajectory of the particle is a straight line with a constant slope v₀ and y intercept x₀. The position-time and velocity-time graphs are shown below.

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  • Uniformly Accelerated Rectilinear Motion Definition: If the acceleration is constant at a value a₀ i.e.  \[a = {a_o}\] 

\[\frac{{dv}}{{dt}} = {a_o}\]

The initial velocity and displacement are \[{v_o}{\text{ and }}{x_o}\] respectively i.e. 

\[v\left( {t = 0} \right) = {v_o}x\left( {t = 0} \right) = {x_o}\]

Integrating the last equation,

\[\int_{{v_o}}^v {dv}  = {a_o}\int_0^t {dt} \]

\[v\left( t \right) = {v_o} + {a_o}t\]

So, velocity varies linearly with time if the acceleration is constant.

Substituting \[v = \frac{{dx}}{{dt}}\]in the expression of v,

\[\frac{{dx}}{{dt}} = {v_0} + {a_0}t\]

Performing integration on both sides,

\[\int_{{x_0}}^x {dx}  = {a_0}\int_0^t {tdt}  + {v_0}\int_0^t {dt} \]

\[x\left( t \right) = {x_0} + {v_0}t + \frac{1}{2}{a_0}{t^2}\]

For constant acceleration, the expression of displacement is quadratic in time.

Time can be eliminated from the expressions of velocity and displacement by substituting \[t = \frac{{v - {v_0}}}{{{a_0}}}\] in the expression of displacement,

\[x\left( t \right) = {x_0} + {v_0}\left( {\frac{{v - {v_0}}}{{{a_0}}}} \right) + \frac{1}{2}{a_0}{\left( {\frac{{v - {v_0}}}{{{a_0}}}} \right)^2}\]

\[x = {x_0} + \left( {\frac{{v - {v_0}}}{{{a_0}}}} \right)\left( {{v_0} + \frac{{v - {v_o}}}{2}} \right)\]

\[x = {x_0} + \frac{{{v^2} - v_0^2}}{{2{a_0}}}\]

\[{v^2} = v_0^2 + 2{a_0}\left( {x - {x_0}} \right)\]

This equation relates the position and velocity at any arbitrary instant. Since acceleration is constant in time, it can be represented as a straight line parallel to the time axis. Velocity is also linear, but it varies with time so that it is a straight line with a nonzero slope. Displacement is quadratic in time and the trajectory is parabolic.

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  • Motion with Non Uniform Acceleration: Acceleration changes with time and position in these motions. Simple harmonic motion is an example where the magnitude of the acceleration is proportional to position. The trajectory of an SHM is sinusoidal. 

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Example

Free fall under gravity: If the gravitational acceleration \[g\]of an object due to the Earth’s gravitational attraction is considered to be constant over the distance of interest, free fall of an object in the gravitational field of Earth can be approximated as a rectilinear motion with constant acceleration. Any nonconservative force like air resistance, viscosity is considered to be absent in the problem.

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If an object falls freely from a height h above the ground under gravity, its initial height d(t = 0) is h and initial velocity v(t = 0) is zero. The constant acceleration is g = 9.8 m/s2 . Using the expressions of position and velocity,

Velocity at any instant t is,

\[v\left( t \right) = gt\]

Displacement at any instant t is,

\[d\left( t \right) = \frac{1}{2}g{t^2}\]

This displacement from initial height is downwards such that the height of the object decreases with time.

Did you know?

  • The motion of two particles under the action of a central force (e.g. electrostatic force) can be approximated as a rectilinear motion.

  • Free-fall under the Earth’s gravitational field is not actually a rectilinear motion because of the rotation of the Earth. The Coriolis force, due to the rotation, causes the free-fall trajectory to bend.

  • Linear motion and rotation (on a plane) about an axis have similar dynamics. 

FAQ (Frequently Asked Questions)

1. What is Rectilinear Motion? Give Examples.

Rectilinear motion is a particle’s motion along a straight line. The system has one degree of freedom such that only one coordinate is sufficient to analyze the motion. Some examples of rectilinear motion are the movement of a train along a straight railway track, a car’s motion along a straight street, ideal free fall under gravity, the motion of a body suspended to a spring, a lift’s vertical motion etc.

2. What are the Categories of Rectilinear Motion?

A Rectilinear Motion has Three Types: 

  • Uniform motion, which is the motion of a body with zero acceleration. The net force acting on the body is zero. 

  • Uniformly accelerated motion, which is the motion with non-zero constant acceleration i.e. the net force on the system is constant.

  • Motion with non-uniform acceleration. The force, acting on the system, is variable.