गति — संख्याओं और ग्राफ़ों से उसका वर्णन
Distance, displacement, speed and velocity
| Quantity | Kind | Meaning | SI unit |
|---|---|---|---|
| distance | scalar | the length of the actual path | m |
| displacement | vector | the shortest straight line from start to finish, with direction | m |
| speed | scalar | distance ÷ time | m/s |
| velocity | vector | displacement ÷ time | m/s |
The trap: average speed is not the average of the speeds — a car going 30 km/h one way and 60 km/h back over the same distance averages 2 × 30 × 60 ÷ (30 + 60) = 40 km/h, not 45.
गति — संख्याओं और ग्राफ़ों से उसका वर्णन
Acceleration and the three equations of motion
Acceleration a = (v − u) ÷ t, the rate of change of velocity, in m/s². It is negative (retardation) when the body slows down.
For uniform acceleration, with u the initial velocity, v the final velocity, a the acceleration, t the time and s the displacement:
| Equation | Relates |
|---|---|
| v = u + at | velocity and time |
| s = ut + ½at² | position and time |
| v² = u² + 2as | velocity and position |
All three can be read off a velocity–time graph: the slope gives a, and the area under the line gives s.
Worked example — a car starts from rest and reaches 20 m/s in 5 s.
Stopping — a train moving at 20 m/s brakes with a retardation of 2 m/s². It stops after t = 20 ÷ 2 = 10 s and travels s = 20² ÷ (2 × 2) = 100 m.
Free fall — near the Earth, a body falls with a ≈ 9.8 m/s² (often taken as 10 for quick sums), whatever its mass, if air resistance is ignored.
The trap: 'starts from rest' means u = 0; 'comes to rest' means v = 0 — half the errors in these sums come from putting the zero in the wrong place.
गति — संख्याओं और ग्राफ़ों से उसका वर्णन
Reading motion graphs — and circular motion
Distance–time graph
| Shape | Meaning |
|---|---|
| straight line sloping up | uniform speed; the slope is the speed |
| curve getting steeper | speed increasing |
| horizontal line | the body is at rest |
Velocity–time graph
| Shape | Meaning |
|---|---|
| horizontal line | uniform velocity, zero acceleration |
| straight line sloping up | uniform acceleration; slope = a |
| straight line sloping down | uniform retardation |
| area under the graph | displacement |
A distance–time graph can never slope downward (distance never decreases) and can never be vertical (that would mean infinite speed).
Uniform circular motion — a body moving round a circle at constant speed is still accelerated, because the direction of its velocity changes at every point. Its speed is v = 2πr ÷ t, where r is the radius and t the time for one round.
The trap: 'uniform circular motion' has constant speed but changing velocity — so it is accelerated motion, not uniform motion.
गति — संख्याओं और ग्राफ़ों से उसका वर्णन
Teaching motion — and the errors to catch
Classroom activities
Errors students make
| Error | Correction |
|---|---|
| treating velocity and speed as the same | velocity carries direction; it can change while speed stays fixed |
| taking 'negative acceleration' to mean moving backwards | it means the velocity is decreasing in the chosen positive direction |
| using the equations when acceleration is not uniform | the three equations hold only for uniform acceleration |
| mixing units — km/h with seconds | convert everything to m and s first |
| reading slope as area | slope of v–t = acceleration; area under v–t = displacement |
Reaction time and safety — a driver needs about 0.7 s to react. At 72 km/h (20 m/s) the car covers 14 m before the brakes are even pressed — the reason for speed limits near schools.
The trap: when a body is thrown straight up, at the top its velocity is zero but its acceleration is still g downward — acceleration does not vanish when velocity does.
गति — संख्याओं और ग्राफ़ों से उसका वर्णन
Points to Remember