The core idea
Speed describes how fast distance accumulates; velocity includes direction of displacement; acceleration describes change in velocity. A clear reference direction and consistent time interval make the three quantities distinguishable.
1. Position requires a chosen reference
Choose an origin and a positive direction before assigning signs. Along an east–west road, take the library entrance as position x = 0 and east as positive. Position x is then a signed coordinate, measured in metres. Displacement is final position minus initial position, written Δx, where Δ means “change in”. Distance is the total length of the route actually travelled and cannot be negative. Displacement can be positive, negative or zero. A person seated in a moving bus is at rest relative to the seat but moving relative to a roadside tree. Neither description is contradictory: the reference frame is different.
Sources: NCERT: Motion in a Straight Line ↗ · NCERT: Describing Motion Around Us ↗
2. Average speed and average velocity use different numerators
Average speed equals total distance divided by total elapsed time. Average velocity equals displacement divided by that same time. Include waiting time if the question asks about the whole journey. Both quantities use m/s, but velocity needs direction or a sign. To convert km/h to m/s, multiply by 1000/3600, or 5/18, because one kilometre is 1000 metres and one hour is 3600 seconds. Do not simply average two reported speeds unless the corresponding time intervals are equal. A slow section can occupy more time than a fast section, so journey averages must be rebuilt from distances and times rather than guessed from the speed labels.
Sources: NCERT: Motion in a Straight Line ↗ · NCERT: Units and Measurements ↗ · NCERT: Describing Motion Around Us ↗
3. Motion at an instant can differ from the journey average
An instantaneous speed describes motion at a particular moment; a speedometer approximates this, rather than displaying the average since the trip began. Instantaneous velocity also specifies direction. When a cycle follows a curve at constant speed, its velocity changes because direction changes. This is why scalar and vector quantities differ: a scalar such as speed needs magnitude alone, while a vector such as velocity needs magnitude and direction. In straight-line questions a plus or minus sign can represent direction. The size of a negative velocity is still a positive speed. A sign is information about your coordinate choice, not a judgement that the motion is wrong.
Sources: NCERT: Motion in a Straight Line ↗ · NCERT: Describing Motion Around Us ↗
4. Acceleration measures velocity change
Average acceleration is a = (v − u)/t, where u is initial velocity, v final velocity and t the elapsed time. Its unit is m/s²: velocity changes by so many metres per second during each second. For constant acceleration, equal time intervals have equal velocity changes. Negative acceleration means acceleration points in the chosen negative direction; it does not always mean slowing. If velocity and acceleration have the same sign, speed increases. If their signs differ, speed decreases until a possible reversal. Acceleration can exist at an instant when velocity is zero, such as the turning instant in an ideal vertical throw; zero velocity alone does not imply zero force or acceleration.
Sources: NCERT: Motion in a Straight Line ↗ · NCERT: Describing Motion Around Us ↗
5. A graph is a statement about two quantities
On a position–time graph, a line joining two points gives average velocity from its slope. The tangent slope at a point gives instantaneous velocity; on a straight segment these agree. A horizontal segment means position stays fixed, not that the object moves along a flat road. On a velocity–time graph, slope represents acceleration; a horizontal segment means constant velocity. Signed area under a velocity–time graph gives displacement: areas below the time axis count negatively. Total distance uses the absolute contributions instead. For constant acceleration, v = u + at and displacement s = ut + ½at². These follow from the straight-line velocity graph and its average velocity (u + v)/2; that simple average is not valid for every changing-velocity journey.
Sources: NCERT: Motion in a Straight Line ↗ · NCERT: Describing Motion Around Us ↗
6. Worked example: a library errand
Illustrative journey: Asha walks 300 m east from a school gate, then 180 m west, taking 8 minutes altogether including a short stop. Choose east positive. Distance is 300 + 180 = 480 m; displacement is +300 − 180 = +120 m. Time is 8 × 60 = 480 s. Average speed is 480/480 = 1.00 m/s; average velocity is 120/480 = +0.25 m/s, or 0.25 m/s east. These different answers describe different aspects of the same trip. The stop affects both averages because it contributes time without distance or displacement. Neither average tells us the speed at every moment of the walk.
Sources: NCERT: Motion in a Straight Line ↗ · NCERT: Describing Motion Around Us ↗
7. Worked example: a uniformly changing velocity
Illustrative straight-road bus model: velocity increases from 2 m/s to 8 m/s in 6 s, with constant acceleration and no direction reversal. Acceleration is (8 − 2)/6 = 1 m/s². Average velocity is (2 + 8)/2 = 5 m/s, so displacement is 5 × 6 = 30 m. Check with the equation: s = 2 × 6 + ½ × 1 × 6² = 12 + 18 = 30 m. On the velocity graph, the same area is a 12 m rectangle plus an 18 m triangle. If acceleration had varied, the endpoint velocities alone would not justify this area or the simple average.
Read motion from area and slope
Sources: NCERT: Motion in a Straight Line ↗ · NCERT: Describing Motion Around Us ↗
PUT IT INTO PRACTICE
Apply it and check your reasoning
- For an illustrative trolley starting from rest, use constant acceleration 0.5 m/s² for 8 s. Choose its direction of motion positive and calculate final velocity.
- Draw its velocity–time graph and calculate the area under it. Then check displacement using s = ut + ½at².
- Check: final velocity 4 m/s and displacement 16 m. Explain why the average velocity is 2 m/s, and identify the constant-acceleration assumption that permits it.
Check your understanding
Can average velocity be zero after a moving journey?
Yes. Returning to the starting position makes total displacement zero. Distance and average speed can remain positive because the route was not zero in length.
Why is constant speed around a bend accelerated motion?
Velocity changes when direction changes. Acceleration measures the full velocity change, so unchanged speed alone does not establish zero acceleration.
Why can a negative acceleration increase speed?
If the object already moves in the negative direction, acceleration in that direction increases the magnitude of its negative velocity. The sign convention must be stated first.
Why does a horizontal velocity graph differ from a horizontal position graph?
The vertical axes show different quantities. Constant velocity can describe motion, whereas constant position describes rest relative to the chosen frame.
Why should the two speeds of unequal-duration legs not simply be averaged?
Each speed contributes over a different time. Total distance divided by total time weights those contributions correctly; an unweighted average generally does not.