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 the 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 of 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 Δt, the displacement is given by,
x = xf - xi
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,
\[a = \frac{d\upsilon}{dt} = \frac{d^2x}{dt^2}\]
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 categorised 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,
\[\frac{d\upsilon}{dt} = 0\]
\[\upsilon = \upsilon_0\]
\[\upsilon_0\] is the constant velocity. From the above equation,
\[\frac{d\upsilon}{dt} = \upsilon_0\]
\[\int_{x_0}^x dx = \upsilon_0 \int_0^t dt\]
\[x(t) = x_0 + \upsilon_0 t\]
\[x(t) = x_0 + \upsilon_0 t\]
Here, x₀ is the initial position of the particle i.e. x(t=0) = x0
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.
Uniformly Accelerated Rectilinear Motion Definition: If the acceleration is constant at a value a0 i.e. a=a0
\[\frac{d\upsilon}{dt} = a_0\]
The initial velocity and displacement are v0 and x0 respectively i.e.
v(t=0) = v0x (t=0) = x0
Integrating the last equation,
\[\int_{\upsilon_0}^{\upsilon} d\upsilon = a_0 \int_{0}^{t} dt\]
v(t) = v0 + a0t
So, velocity varies linearly with time if the acceleration is constant.
Substituting v = dxdt in the expression of v,
dxdt = v0 + a0t
Performing integration on both sides,
\[\int{x}{x_0} dx = a_0 \int_{0}^{t} tdt + \int_{\upsilon_0}^{\upsilon} d\upsilon \]
\[x(t) = x_0 + \upsilon_0 t + \frac{1}{2} a_0 t^2\]
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=v -v0 a0 in the expression of displacement,
\[x = x_0 + (\frac{\upsilon - \upsilon_0}{a_0}) (\upsilon_0 \frac{\upsilon - \upsilon_0}{2})\]
\[x(t) = x_0 + (\upsilon_0 \frac{\upsilon - \upsilon_0}{a_0}) + \frac{1}{2} a_0( \frac{\upsilon - \upsilon_0}{a_0})^2 \]
\[x = x_0 + \upsilon^2 - \frac{\upsilon_0^2}{2a_0}\]
\[\upsilon^2 = \upsilon_0^2 + 2a_0 (x - x_0) \]
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.
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.
Example
Free fall under gravity: If the gravitational acceleration got 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.
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(t)=gt
Displacement at any instant t is,
d(t)=12gt2
This displacement from the 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.
Places like museums, retail stores, and even buildings require linear motion control.
FAQs on Rectilinear Motion of Particles
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.
3. What is motion? State the different types of motions.
Motion is basically the free movement of a body with respect to time. The different types of motions are classified on the basis of different aspects such as time, distance, path, and speed.
There are four main types of motions and these are as follows.
Rotatory Motion: This type of motion involves the movement of a particular object in a circular motion that takes place around an axis of rotation. A very common yet popularly known example would be the Earth’s rotation.
Oscillatory Motion: This type of motion tends to repeat itself. It is also referred to as periodic motion. A good example would be a child swinging on a swing; the constant swinging would be considered to be an oscillatory motion.
Linear Motion: This motion tends to be one-dimensional along a straight line. It is also called rectilinear motion and with reference to mathematics, it is said to use only one spatial dimension. For instance, an athlete running on a track that is straight.
Reciprocating Motion: As the name might suggest, this is a motion that is continuous and repetitive in either a forward and backward direction or an upward and downward one. The mechanism of an electric doorbell is a good example of the same.
4. What is linear motion? How is it useful in everyday life?
Linear or rectilinear motion is a type of motion that is one-dimensional and takes place along a straight line. The application of linear motion is quite important as it tends to take place in a number of ways in different aspects of life. Some examples of rectilinear motion are as follows.
It is utilised in the medical field in CT scanners, MRI scanners, X-ray machines, and even the beds of operating rooms and dentist chairs.
A lot of furniture items also require linear motion.
In engineering, the application of linear motion is responsible for reducing friction in all kinds of machinery.
5. What are the main characteristics of rectilinear motion? State its types.
The main characteristics of rectilinear motion are as follows:
It does not have a normal pace of acceleration which is how no change of direction takes place in this type of motion. In other words, the force that acts on the system is variable.
The trajectory of this motion is a straight line.
There are three main types of rectilinear motions and they are as follows:
Uniform rectilinear motion wherein the given object travels with zero acceleration and at a constant speed.
Uniformly accelerated rectilinear motion wherein the given object tends to travel with constant acceleration.
Rectilinear motion that has a non-uniform acceleration wherein the given object tends to travel at both an irregular speed as well as irregular acceleration.