When we look at a car's speedometer reading 65 mph, what does that really mean? If the velocity remains constant, we know that after an hour we will have traveled 65 miles. But if the velocity varies, what does it mean to say that the velocity at a given instant is 65 mph?
To understand this concept, we need to explore the relationship between average velocity over a time interval and instantaneous velocity at a specific moment.
Let's examine the motion of a racing car traveling along a straight track. Assume we can measure the distance traveled by the car (in feet) at different time intervals:
| Time (s) | 0 | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|---|
| Distance (ft) | 0 | 3 | 12 | 27 | 48 | 75 |
Let's visualize this data on a distance-time graph:
To find the velocity at t = 2, we'll calculate the average velocity over a time interval [2, 3]:
\[ \text{Average velocity} = \frac{\text{change in position}}{\text{time elapsed}} = \frac{f(3) - f(2)}{3 - 2} = \frac{27 - 12}{1} = 15 \text{ ft/s} \]
Let's examine what happens when we calculate average velocities over smaller and smaller time intervals. Here's data at finer 0.1-second intervals near t = 2:
| Time (s) | 2.0 | 2.1 | 2.2 | 2.3 | 2.4 | 2.5 |
|---|---|---|---|---|---|---|
| Distance (ft) | 12.00 | 13.23 | 14.52 | 15.87 | 17.28 | 18.75 |
| Time interval | [2, 3] | [2, 2.5] | [2, 2.4] | [2, 2.3] | [2, 2.2] | [2, 2.1] |
|---|---|---|---|---|---|---|
| Average velocity (ft/s) | 15.0 | 13.5 | 13.2 | 12.9 | 12.6 | 12.3 |
As we take smaller and smaller intervals, the average velocities appear to be approaching 12 ft/s. This is the instantaneous velocity at exactly t = 2.
The instantaneous velocity at time t = a is defined as:
\[ v = \lim_{\Delta t \to 0} \frac{f(a + \Delta t) - f(a)}{\Delta t} \]
or equivalently:
\[ v = \lim_{t \to a} \frac{f(t) - f(a)}{t - a} \]
This is the same expression as the slope of the tangent line to the distance-time curve at the point where t = a.
Use the controls to explore how average velocity approaches instantaneous velocity as the interval gets smaller:
Average velocity over [2, 3]:
\[ \frac{f(3) - f(2)}{3 - 2} = \frac{27 - 12}{1} = 15 \text{ ft/s} \]
As Δt approaches 0, this approaches the instantaneous velocity at t = 2:
\[ v = 12 \text{ ft/s} \]
The concept of instantaneous velocity is fundamental in:
In each case, we're interested in the rate of change at a specific moment, not just the average over a time period.