What does a speedometer actually measure?
It shows your speed at that instant, which is the magnitude of the instantaneous velocity. It is not your average over the trip. In physics, instantaneous velocity is the limit of Δx/Δt as the time interval Δt shrinks towards zero, written v = dx/dt. The speedometer drops the sign and shows only the size.
Average velocity hides the details
Say a bus covers 60 km in 2 hours. Its average speed is 30 km/h. Did the needle sit on 30 the whole way? It stopped at signals, crawled through a market, then ran at 50 on the open road. The average tells you nothing about any single moment.
NCERT starts Chapter 2 by calling motion a "change in position of an object with time". It also treats the moving thing as a point object, which is fair when its size is much smaller than the distance it covers in a reasonable time. What we want now is the velocity at one particular instant, say exactly 4 seconds after the start.
Shrink the interval and watch the number settle
Here is how the book does it. Take a car whose position is x = 0.08 t³, with x in metres and t in seconds. We want its velocity at t = 4 s.
Pick an interval Δt centred on 4 s, so it runs from 4 − Δt/2 to 4 + Δt/2. Find x at both ends, then divide Δx by Δt. Table 2.1 does this again and again with smaller intervals:
| Δt (s) | Δx/Δt (m s⁻¹) |
|---|---|
| 2.0 | 3.92 |
| 1.0 | 3.86 |
| 0.5 | 3.845 |
| 0.1 | 3.8402 |
| 0.01 | 3.8400 |
By Δt = 0.1 s the value reads 3.8402, and at 0.01 s it is 3.8400. It has settled. So the velocity at t = 4 s is 3.84 m s⁻¹.
We never put Δt = 0. That would give 0/0, which means nothing. Δt gets as small as we like and the ratio heads to a fixed value. That fixed value is the limit:
v = lim (Δt → 0) Δx/Δt = dx/dt
On the x–t graph this has a neat picture. Each Δx/Δt is the slope of a chord joining two points on the curve. As Δt shrinks, the two points slide together and the chord turns into the tangent at t = 4 s. The slope of that tangent is the instantaneous velocity.
Calculus does it in one line
Example 2.1 shows why dx/dt saves time. A particle moves along the x-axis with x = a + bt², where a = 8.5 m and b = 2.5 m s⁻².
Differentiate: v = dx/dt = 2bt = 5.0 t m s⁻¹. The constant a vanishes, because a starting position doesn't change how fast you move. So:
- at t = 0, v = 0
- at t = 2.0 s, v = 10 m s⁻¹
The same example also asks for the average velocity from t = 2.0 s to 4.0 s. That is [x(4) − x(2)] / 2.0 = 6.0b = 15 m s⁻¹. An average belongs to an interval. An instantaneous value belongs to a moment.
If an object moves with uniform velocity, the instantaneous velocity equals the average velocity at every moment.
Speed is the size, velocity carries the sign
On a straight line, velocity has a sign. A car moving at +24 m s⁻¹ and another at −24 m s⁻¹ go in opposite directions, yet both have a speed of 24 m s⁻¹. Instantaneous speed is just the magnitude of instantaneous velocity.
That's what the speedometer shows. It reads 60 whether you drive north or south. It can't tell direction, so it measures speed, not velocity.
Averages behave differently. Average speed uses the total path length. Average velocity uses the change in position. If you walk 5 m forward and 5 m back, your average velocity is zero but your average speed isn't. Average speed is always at least the magnitude of average velocity. The two are equal only when the object never turns back along its straight line.
Check yourself
For x = 8.5 + 2.5t² (in metres), what is the instantaneous velocity at t = 1.5 s? (Answer at the bottom.)
Questions students ask
What is instantaneous velocity?
It is the velocity at one instant: the limit of Δx/Δt as Δt tends to zero, written dx/dt. On an x–t graph it equals the slope of the tangent at that instant.
Does a speedometer show speed or velocity?
Speed. It gives the magnitude of the instantaneous velocity and says nothing about direction.
Why can't we just put Δt = 0?
Then Δx is also zero and Δx/Δt becomes 0/0, which has no value. We take the limit as Δt gets smaller instead.
Is instantaneous speed always equal to the magnitude of instantaneous velocity?
Yes. Average speed and average velocity can differ, but at a single instant the speed is exactly the size of the velocity.
When does average velocity equal instantaneous velocity?
In uniform motion. If the velocity never changes, every instant has the same velocity as the average.
Answer: v = 2bt = 5.0 × 1.5 = 7.5 m s⁻¹.
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