Motion in a Straight Line: Distance, Velocity, Acceleration & the Equations of Motion
A friendly, from-scratch guide to kinematics in one dimension: distance vs displacement, speed vs velocity, acceleration, and the three equations of uniformly accelerated motion, with a full worked example and a clear derivation. Aligned to the NCERT Class 11 syllabus and the foundation for JEE, NEET and CUET.
By the PadhoDost Team · 📖 7 min read · Updated 4 August 2026
Part of Class 11 (CBSE) prep🧠 The auto-rickshaw meter
Imagine you take an auto from home to your coaching class. The meter charges you for every single metre the auto actually rolls, down your gali, around the sabzi market, past the temple. That whole winding path is the distance. But if you drew one straight arrow from your home to the class, that shortcut arrow is the displacement. On a twisty route the meter might read 3 km even though the class is only 1 km away in a straight line. Same trip, two very different numbers, and that gap is the heart of this whole chapter. Padho, dost, this one clicks fast!
Distance vs Displacement
Distance is the total length of the actual path you travel. It is a scalar; it has size (magnitude) only, no direction, and it can never be negative. Displacement is the straight-line change in position, pointing from where you started to where you ended. It is a vector; it has both size and direction. On a straight line we show that direction with a simple + or - sign. If you walk 300 m east and then 300 m back west to your starting point, your distance is 600 m but your displacement is 0 m. Both quantities are measured in the SI unit metre (m).
Speed vs Velocity
Speed tells you how fast you cover distance; it is a scalar. Average speed = total distance divided by total time. Velocity tells you how fast your position changes AND in which direction; it is a vector. Average velocity = displacement divided by total time. Because velocity uses displacement, it can be zero even on a long journey. If a runner completes one full lap of a 400 m track in 100 s, her average speed is 4 m/s, but her average velocity is 0 m/s because she finishes exactly where she began. Both are measured in metres per second (m/s). 'Instantaneous' speed or velocity is the value at a single instant, exactly what a speedometer reads.
Acceleration: how quickly velocity changes
Acceleration is the rate at which velocity changes with time. It is a vector, measured in metres per second squared (m/s^2). If a bike speeds up from 0 to 10 m/s in 5 s, its velocity changed by 10 m/s over 5 s, so its acceleration is 2 m/s^2. When velocity decreases (braking), the acceleration points opposite to the motion and comes out negative; this is often called retardation or deceleration. 'Uniform acceleration' means the velocity changes by equal amounts in equal time intervals, so the acceleration stays constant throughout.
The Three Equations of Uniformly Accelerated Motion
Where these equations come from
- 1Start from the definition of acceleration: a = (v - u) / t.
- 2Rearrange it: multiply both sides by t to get v - u = at, so v = u + at. That is Equation 1.
- 3For constant acceleration the velocity rises steadily, so the average velocity is simply (u + v)/2. Displacement = average velocity x time, so s = [(u + v)/2] x t.
- 4Substitute v = u + at into that: s = [(u + u + at)/2] x t = [(2u + at)/2] x t = ut + (1/2)at^2. That is Equation 2.
- 5From Equation 1, t = (v - u)/a. Put this into s = [(u + v)/2] x t: s = [(u + v)/2] x [(v - u)/a] = (v^2 - u^2) / (2a).
- 6Rearrange: v^2 - u^2 = 2as, so v^2 = u^2 + 2as. That is Equation 3.
📝 Worked example: a bus pulling away
A bus starts from rest and accelerates uniformly at 2 m/s^2 for 5 s. Find its final velocity and the distance it covers.
List the knowns: u = 0 m/s (starts from rest), a = 2 m/s^2, t = 5 s.
Final velocity, use v = u + at: v = 0 + (2 x 5) = 10 m/s.
Distance, use s = ut + (1/2)at^2: s = (0 x 5) + (1/2) x 2 x (5)^2 = 0 + (1/2) x 2 x 25 = 25 m.
Cross-check with v^2 = u^2 + 2as: right side = 0 + (2 x 2 x 25) = 100, and v^2 = 10^2 = 100. They match, so the answer is consistent.
Answer: final velocity = 10 m/s, distance covered = 25 m.
| Quantity | Scalar or Vector | What it measures | Can it be negative? | SI unit |
|---|---|---|---|---|
| Distance | Scalar | Total path length travelled | No (always >= 0) | metre (m) |
| Displacement | Vector | Straight-line change in position | Yes | metre (m) |
| Speed | Scalar | Distance / time | No (always >= 0) | m/s |
| Velocity | Vector | Displacement / time | Yes | m/s |
| Acceleration | Vector | Change in velocity / time | Yes | m/s^2 |
Quick revision
- ✓Distance is the total path (scalar, never negative); displacement is the straight-line change in position (vector, can be zero or negative).
- ✓Average speed = distance / time; average velocity = displacement / time. Velocity can be zero even when speed is not.
- ✓Acceleration = (v - u) / t, measured in m/s^2; a negative value means the body is slowing down.
- ✓For constant acceleration: v = u + at, s = ut + (1/2)at^2, and v^2 = u^2 + 2as.
- ✓Always choose a positive direction first, then keep your signs and SI units consistent throughout the problem.
⚡ Quick check
A student jogs one full round of a 400 m circular track and stops exactly where she started, taking 200 s. What is her average velocity for the round?
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