Rectilinear Motion (Motion in a Straight Line)

 

Rectilinear motion is just a fancy physics term for "moving in a straight line." Whether it’s an ant crawling along the edge of a ruler or a jet blasting down a runway, if it’s not turning, it’s rectilinear.

For exams like NEET and JEE, this is the "bread and butter" topic. It’s where you learn to juggle displacement, velocity, and acceleration.

Rectilinear motion is one of the most fundamental topics in kinematics and forms the base of many NEET, JEE Main, and JEE Advanced problems. Almost every concept of motion—velocity, acceleration, graphs, equations of motion—can be understood first through rectilinear motion.

The Basics: Tools of the Trade

Before we run, we need to walk. Here are the parameters you need to master:

• Distance vs. Displacement: Distance is the total path covered (scalar). Displacement is the shortest straight-line gap between start and finish (vector).
• Speed vs. Velocity: Speed is how fast; Velocity is how fast plus in what direction.
• Acceleration (a): The rate at which velocity changes. If you speed up, slow down, or change direction, you’re accelerating.

Case Study: The "U-Turn" Trap

Imagine a particle moves from x=0 to x=10m and then back to x=4m.

• Distance: 10+6=16m.
• Displacement: Final position (4) - Initial position (0) = 4m.

Note: In JEE, always check if the velocity becomes zero and reverses. This is where distance and displacement diverge.

1.What is Rectilinear Motion?

Definition

Rectilinear motion is the motion of a particle along a straight-line path.

The word comes from two Latin words:

• Rectus → straight
• Linea → line

So rectilinear motion literally means straight-line motion.

✔ Examples in Daily Life

• A train moving on a straight railway track
• A car moving on a straight highway
• A stone falling vertically
• An elevator moving up or down

In all these cases, motion occurs only along one straight line.

2.Characteristics of Rectilinear Motion

Rectilinear motion has some important features:

I. Motion occurs along one dimension (1D motion)

II.Position of the particle can be described using one coordinate (x-axis)

III.Velocity and acceleration occur along the same line

IV.Direction is usually described as:

• Positive direction (+x)
• Negative direction (−x)

3.Types of Rectilinear Motion

The Three Flavours of Rectilinear Motion

Rectilinear motion can occur in different ways depending on velocity and acceleration.

I. Uniform Rectilinear Motion

Definition

If a particle moves in a straight line with constant velocity, it is called uniform rectilinear motion.

Velocity is constant (a=0). No speeding up, no slowing down.

·        Formula: Displacement =Velocity×Time

v = constant

Acceleration:

a = 0

Example

A train moving at 60 km/h on a straight track.

• Velocity remains constant
• No acceleration

Position changes linearly with time.

Position Equation

x = x₀ + vt

where

(x₀) = initial position
(v) = constant velocity
(t) = time

II. Uniformly Accelerated Motion (UAM)

This is the "Superstar" of NEET/JEE. Acceleration is constant. We use the Equations of Motion:

1.    v=u+at

2.    s=ut+1​/2at²

3.    v²=u²+2as

4.    s_n​=u+a/2​(2n−1) (Displacement in the nth second)

III.Non-Uniform Rectilinear Motion

When velocity changes with time, motion becomes non-uniform rectilinear motion.

Acceleration is a function of time, position, or velocity (e.g., a=3t2).

·        Method: Use Calculus.

o   To go "Forward" (Position → Velocity → Acceleration): Differentiate.

o   To go "Backward" (Acceleration → Velocity → Position): Integrate.

Acceleration exists.

Example:

• A car accelerating from rest
• A falling stone

4.Equations of Rectilinear Motion

For uniform acceleration, three important equations are used.

First Equation of Motion

v = u + at

where

u = initial velocity
v = final velocity
a = acceleration
t = time

Second Equation of Motion

s = ut + 1/2at²

Third Equation of Motion

v² = u² + 2as

These equations are used in most NEET and JEE problems.

5.Graphical Representation

Rectilinear motion can also be analysed using graphs.

Position–Time Graph

Slope of graph → velocity

Cases:

• Straight line → constant velocity
• Horizontal line → object at rest
• Curve → changing velocity

Velocity–Time Graph

Slope → acceleration

Area under graph → displacement

6.Case Study 1: Elevator Motion

Consider a lift moving from ground floor to the 5th floor.

Stages of motion:

1️⃣ Lift starts from rest
2️⃣ It accelerates upward
3️⃣ Moves with constant velocity
4️⃣ Slows down before stopping

This is rectilinear motion with changing acceleration.

7.Case Study 2: Free Fall of a Stone

When a stone is dropped from a height:

• Motion occurs vertically downward
• Acceleration due to gravity

g = 9.8 m/s²

Velocity increases every second.

Example:

Time	Velocity
1 s	9.8 m/s
2 s	19.6 m/s
3 s	29.4 m/s

This is uniformly accelerated rectilinear motion.

8.Case Study 3: Motion of a Car on Highway

A car on a straight highway experiences different types of rectilinear motion:

Stage 1: Starting → acceleration
Stage 2: Cruise → constant velocity
Stage 3: Braking → negative acceleration

So rectilinear motion can include:

• acceleration
• constant velocity
• deceleration

Special Case: Motion Under Gravity

When an object is dropped or thrown vertically, it’s still rectilinear motion, just on the Y-axis.

·        Acceleration: g≈9.8 m/s2 (downward).

·        Key Trick: Always pick a sign convention. Usually, Up is Positive (+) and Down is Negative (-).

Example: If you throw a ball up at 20 m/s, at its highest point, v=0. Using v=u+at: 0=20+(−10)t⟹t=2 seconds to reach the top.

9.Numerical Problems

Numerical 1 (NEET Level)

A car starts from rest and accelerates at 2 m/s² for 5 seconds.

Find final velocity.

v = u + at

v = 0 + 2 x 5

v = 10  m/s

Numerical 2 (JEE Main Level)

A body moves with initial velocity 10 m/s and acceleration 2 m/s² for 4 s.

Find displacement.

s = ut + 1/2at²

s = 10(4) + 1/2(2)(4²)

s = 40 + 16

s = 56  m

Numerical 3 (JEE Advanced Level)

A particle moves along a straight line with acceleration:

a = 4t

Find velocity at t = 3 s, if initial velocity is 2 m/s.

Acceleration relation:

a = dv/dt

dv/dt = 4t

Integrate:

v = 2t² + C

Initial condition:

v=2 when t=0

So (C=2)

v = 2t² + 2

At (t=3)

v = 2(9)+2

v = 20 m/s

The "Brain Teaser" Numerical (JEE Level)

Problem: A police jeep is chasing a thief. The jeep moves at a constant speed of 20 m/s. The thief starts his bike from rest (when the jeep is 100m behind) with a constant acceleration of 2 m/s2. Will the police catch the thief?

Thinking Process:

1.    Let them meet at time t.

2.    Jeep's distance: d_j​=20t (Constant speed).

3.    Thief's distance: d_t​=0(t)+1​/2(2)t²=t² (Starting from rest).

4.    For them to meet: d_j​=d_t​+100 (The jeep has to cover the 100m gap).

5.    Equation: 20t=t²+100⟹t²−20t+100=0.

6.    Solve: (t−10)²=0⟹t=10 seconds.

7.    Answer: Yes, the police catch him in 10 seconds.

10.Important Concepts for Exams

Key Points

✔ Motion occurs along one straight line
✔ Only one coordinate is needed
✔ Velocity and acceleration are collinear

Important Results

If velocity is constant:

Acceleration is defined as the rate of change of velocity over time. Mathematically, this is expressed as:

 

a=dv/dt

If velocity is constant, its derivative with respect to time is zero; therefore, the acceleration must be zero

a=0
If acceleration is constant:

v = u + at

It represents the first equation of motion for an object moving with a constant acceleration. This formula is derived by integrating the constant acceleration a with respect to time:

In this equation, v is the final velocity, u is the initial velocity, a is the constant acceleration, and

t is the time elapsed. This linear relationship indicates that for every unit of time, the velocity increases by a fixed amount a.

⭐ Quick Revision Table

Quantity	Formula
Velocity	(v = dx/dt)
Acceleration	(a = dv/dt)
First equation	(v = u + at)
Second equation	(s = ut + 1/2at2)
Third equation	(v2 = u2 + 2as)

How to attack these in exams:

• NEET: Focus on direct application of v²−u²=2as and Motion Under Gravity. They love "ratio" questions (e.g., ratio of distances in successive seconds).
• JEE Main: Focus on Graphs (Slope of x−t is velocity, Area under v−t is displacement).
• JEE Advance: Focus on Variable Acceleration and Relative Motion in 1D.

Final Idea

Rectilinear motion is the simplest form of motion where a particle moves along a straight path and its position changes only along one axis.

Understanding rectilinear motion makes it easier to learn:

• Projectile motion
• Circular motion
• Relative motion
• Advanced mechanics