Rectilinear Motion Formulas, Equations & Solved Problems
Rectilinear motion refers to the movement of a particle or object along a straight line. In this type of motion, the position, velocity, and acceleration are all restricted to one dimension, making calculations and analysis more straightforward. Objects moving in a straight path, like a car along a straight road or a stone dropped vertically, are typical examples of rectilinear motion.
The study of rectilinear motion forms the basis of many advanced topics in physics, especially in kinematics and mechanics. Understanding these basics helps in solving complex numerical problems and builds a strong conceptual foundation.
Rectilinear Motion: Key Concepts
Rectilinear motion can be classified based on how the velocity or acceleration behaves:
- Uniform Rectilinear Motion: The object moves with a constant velocity (zero acceleration).
- Uniformly Accelerated Rectilinear Motion: The object moves with a constant, non-zero acceleration.
- Non-uniformly Accelerated Motion: The acceleration is variable and changes with time or position.
Position, Displacement, and Distance
The position of a particle is its location along the straight line (say, x-axis), measured from a reference point (origin).
Displacement is the difference between the final and initial positions:
Distance is the actual path covered, while displacement is the shortest path (straight line) between two points. Displacement can be positive or negative based on direction, but distance is always positive.
Speed, Velocity, and Acceleration
Speed is the rate at which distance is covered (scalar), while velocity is the rate of change of displacement (vector):
Acceleration is the rate at which velocity changes with time:
In rectilinear motion, it is often enough to work with magnitudes, as the direction is along the single chosen axis.
Types of Rectilinear Motion with Equations
For constant acceleration, three fundamental equations apply:
- v = u + at
- s = ut + ½ at²
- v² = u² + 2as
Where:
v = final velocity
u = initial velocity
a = acceleration
t = time
s = displacement
| Equation | Usage | SI Units |
|---|---|---|
| v = u + at | Finds velocity after time t | m/s |
| s = ut + ½ at² | Finds displacement after time t | m |
| v² = u² + 2as | Relates velocity and displacement (time independent) | m²/s² |
Solved Example: Free Fall Under Gravity
Suppose a stone is dropped from rest from a height of 20 m.
Given: u = 0, s = 20 m, a = g = 9.8 m/s²
Using v² = u² + 2as:
v = √392 ≈ 19.8 m/s
The velocity just before hitting the ground is approximately 19.8 m/s.
| Aspect | Rectilinear Motion | Curvilinear Motion |
|---|---|---|
| Path | Straight line | Curved/Circular |
| Key Feature | One-dimensional | Two or three-dimensional |
| Examples | Car on highway, stone dropped vertically | Projectile, planetary orbits |
Problem-Solving Approach
| Step | Description |
|---|---|
| 1 | List all known values with units. |
| 2 | Identify what needs to be found (e.g., velocity, time). |
| 3 | Select the correct equation of motion. |
| 4 | Substitute the values into the formula. |
| 5 | Calculate step by step. |
| 6 | Write the final answer with proper units. |
Application Examples in Daily Life
- A car moving on a straight road.
- A train on a straight track.
- An elevator going up or down in a shaft.
- An apple falling vertically (neglecting air resistance).
Graphs in Rectilinear Motion
A position-time graph can indicate whether an object is at rest, moving at constant velocity, or accelerating.
A velocity-time graph helps in visually understanding the nature of motion—constant velocity (horizontal line), uniform acceleration (straight line with a slope).
Explore More and Practice
- Read about derivation of motion equations in detail.
- Practice with uniformly accelerated motion problems.
- Concepts like distance vs displacement and speed vs velocity are important for clarity.
- Use velocity and acceleration links for deeper understanding.
- Learn more about graphical representation of motion for exams and board tests.
Next Steps
- Strengthen your basics with stepwise problem-solving and visual aids.
- Practice regularly with topic-wise questions available in Vedantu resources.
- Explore related topics such as motion in a straight line and uniform vs non-uniform motion.
FAQs on Rectilinear Motion of Particles Explained for Students
1. What is rectilinear motion and what are its key characteristics?
Rectilinear motion is the movement of an object or particle along a straight line.
Key characteristics:
- It is a one-dimensional motion.
- The object’s path does not deviate from a straight line.
- Only one spatial coordinate is needed for analysis.
- Displacement, velocity, and acceleration vectors all point along the same line.
- Examples include cars moving on straight roads, objects in free fall (ignoring air resistance), and elevators moving vertically.
2. What are the different types of rectilinear motion based on acceleration?
Rectilinear motion can be classified based on acceleration:
1. Uniform rectilinear motion: Zero acceleration, constant velocity.
2. Uniformly accelerated rectilinear motion: Constant, non-zero acceleration (e.g., free fall under gravity).
3. Non-uniformly accelerated rectilinear motion: Acceleration varies with time or position.
Each type uses different equations or calculus methods to describe motion.
3. Can you provide some real-world examples of rectilinear motion?
Common examples of rectilinear motion include:
- A car moving in a straight lane on a highway.
- A train running on a straight railway track.
- An apple falling vertically from a tree (when ignoring air resistance).
- An elevator moving up or down a vertical shaft.
- A ball dropped from a rooftop.
In all these cases, the path followed is a straight line and the analysis can be done using rectilinear motion equations.
4. How does rectilinear motion differ from curvilinear motion?
The main difference lies in the path of motion:
- Rectilinear motion: Movement occurs strictly along a straight line (one-dimensional).
- Curvilinear motion: Movement follows a curved or circular path (two or three-dimensional).
- In rectilinear motion, velocity direction does not change. In curvilinear motion, direction of motion continuously changes.
- Rectilinear motion is analyzed with simpler, one-dimensional equations; curvilinear motion requires vector analysis in multiple dimensions.
5. What are the fundamental equations of motion for uniformly accelerated rectilinear motion?
The three fundamental equations for rectilinear motion with constant acceleration are:
1. v = u + at
2. s = ut + ½ at²
3. v² = u² + 2as
Where:
- u = initial velocity
- v = final velocity
- a = acceleration
- t = time
- s = displacement
These equations help solve most problems involving uniform acceleration along a straight line.
6. Why are the three standard equations of motion not applicable to all types of rectilinear motion?
These equations are valid only if acceleration is constant throughout the motion.
If acceleration changes with time or position (non-uniform), these formulas cannot be used directly.
In non-uniform acceleration:
- Calculus-based methods (integration and differentiation) are needed to relate displacement, velocity, and acceleration.
This ensures accurate calculation for complex or variable motion cases.
7. Is it possible for a particle in rectilinear motion to have zero velocity but still be accelerating?
Yes, this is possible.
- Acceleration is the rate of change of velocity, not velocity itself.
- For example: When an object is thrown vertically upwards, at the highest point, its instantaneous velocity is zero, but acceleration due to gravity still acts downward.
- Therefore, an object can have zero velocity at an instant and still be experiencing non-zero acceleration.
8. How are position-time and velocity-time graphs used to analyse rectilinear motion?
Graphs help visualize and analyse motion:
- Position-time (x–t) graph:
The slope at any point gives the instantaneous velocity.
Straight line: Uniform motion.
Curve: Indicates acceleration.
- Velocity-time (v–t) graph:
The slope gives acceleration.
The area under the curve equals displacement.
Graphs make it easier to interpret complex motion behaviours and identify acceleration patterns.
9. What is the difference between displacement and distance in rectilinear motion?
Distance is the total length of the path travelled, regardless of direction.
Displacement is the shortest straight-line distance from the initial to the final position, along the direction of motion.
Key Points:
- Distance is a scalar, displacement is a vector.
- Displacement can be zero even if distance is not (when returning to start point).
10. What is meant by uniform and non-uniform rectilinear motion?
Uniform rectilinear motion means the object moves in a straight line with constant velocity (zero acceleration).
Non-uniform rectilinear motion means the object’s velocity changes with time (acceleration is not zero).
Summary:
- In uniform motion, equal distances are covered in equal time intervals.
- In non-uniform motion, distances covered in equal intervals are unequal due to varying velocity or acceleration.
11. What are common mistakes students make in rectilinear motion questions?
Common mistakes include:
- Confusing distance with displacement.
- Incorrect unit conversions (e.g., km/h to m/s).
- Applying uniform acceleration formulas to non-uniform scenarios.
- Not using the correct equation for the quantities given.
- Ignoring direction (sign convention) in vector quantities.
To avoid these, always check units, signs, and the type of motion described.
12. Can you list high-frequency exam questions from rectilinear motion?
High-frequency exam question types include:
- Numerical problems on equations of motion (v = u + at, etc.).
- Graph interpretation (x-t and v-t graphs).
- Identifying scalar and vector quantities (distance, displacement, velocity, speed, acceleration).
- Real-world application scenarios (e.g., free fall, vertical throw).
- Differences between rectilinear and curvilinear motion.
Practicing these helps in scoring well in competitive exams and board tests.
