Introduction
Motion in a straight line (also called rectilinear motion or 1D motion) is the simplest form of motion where an object moves along a single axis.
Distance vs Displacement
Understanding the difference between these two is crucial:
| Property | Distance | Displacement |
|---|---|---|
| Definition | Total path length traveled | Change in position |
| Nature | Scalar | Vector |
| Sign | Always positive | Can be +, -, or 0 |
| Formula | Sum of all path lengths | $\vec{s} = \vec{r}_f - \vec{r}_i$ |
Visual Example
Start (A) ----5m----> (B) ----3m----> (C)
<---2m---- (C) back to (D)
If you walk from A to C and back to D:
- Distance = 5 + 3 + 2 = 10 m
- Displacement = 5 + 3 - 2 = 6 m (in original direction)
Quick Check
Distance ≥ |Displacement| always. They’re equal only when motion is in one direction without any reversal.
Speed vs Velocity
Speed (Scalar)
Speed is the rate at which distance is covered.
$$\text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}}$$- Always positive
- SI Unit: m/s
Velocity (Vector)
Velocity is the rate of change of displacement.
$$\text{Average Velocity} = \frac{\text{Displacement}}{\text{Time}} = \frac{\Delta \vec{s}}{\Delta t}$$ $$\text{Instantaneous Velocity} = \lim_{\Delta t \to 0} \frac{\Delta \vec{s}}{\Delta t} = \frac{d\vec{s}}{dt}$$- Can be positive, negative, or zero
- SI Unit: m/s
Important Relationships
| Situation | Speed vs Velocity |
|---|---|
| Uniform motion in one direction | Speed = |
| Motion with reversal | Speed > |
| Complete round trip | Speed > 0, Average Velocity = 0 |
Acceleration
Acceleration is the rate of change of velocity.
$$\vec{a} = \frac{d\vec{v}}{dt}$$Interactive Demo: Visualize Velocity and Acceleration
Explore how velocity and acceleration change with time in 1D motion.
Types of Acceleration
- Positive Acceleration: Velocity increasing in positive direction
- Negative Acceleration: Velocity decreasing (also called deceleration or retardation)
- Zero Acceleration: Constant velocity (uniform motion)
Common Misconception
Negative acceleration doesn’t always mean slowing down!
- A car moving left (negative direction) with negative acceleration is actually speeding up.
- What matters is whether $\vec{a}$ and $\vec{v}$ are in the same direction.
Acceleration Direction Rules
| $\vec{v}$ and $\vec{a}$ | Motion |
|---|---|
| Same direction | Speeding up |
| Opposite directions | Slowing down |
Summary Table
| Quantity | Symbol | Type | Formula | SI Unit |
|---|---|---|---|---|
| Distance | $d$ | Scalar | Total path | m |
| Displacement | $\vec{s}$ | Vector | $\vec{r}_f - \vec{r}_i$ | m |
| Speed | $v$ | Scalar | $\frac{d}{t}$ | m/s |
| Velocity | $\vec{v}$ | Vector | $\frac{d\vec{s}}{dt}$ | m/s |
| Acceleration | $\vec{a}$ | Vector | $\frac{d\vec{v}}{dt}$ | m/s² |
Practice Problems
Problem 1
A car travels 100 m east, then 60 m west. Find the distance traveled and displacement.
Solution:
- Distance = 100 + 60 = 160 m
- Displacement = 100 - 60 = 40 m east
Problem 2
A particle moves along x-axis. Its position at time $t$ is $x = 3t^2 - 2t + 5$ m. Find velocity and acceleration at $t = 2$ s.
Solution:
- $v = \frac{dx}{dt} = 6t - 2$
- At $t = 2$: $v = 6(2) - 2 = $ 10 m/s
- $a = \frac{dv}{dt} = 6$ m/s² (constant)