Imagine you are travelling in a car on a straight road. You could work out the speed of your car by noting the time taken to travel beween two kilometer signs by the roadside. You could note the time when you pass one kilometer sign and again note the time when you pass the next kilometer sign.Or if the Odometer of car ( The one that indicates the distance travelled- imagine the speedometer part is not working ) could be imagined to show distances to a greater accuracy, you could read off the distance at regular intervals of time. You would then have the option of finding the distance travelled in small intervals of time. All you need is the time taken to travel a certain distance or distance travelled in a certain time to work out the speed.
If the car's speed did not increase or decrease during the time you worked out the speed, the calculated value will also be speed of the car at any instant of motion. What if the speed of the car varied? You would then be getting the average speed for the duration of your measurement. If you could make the distance intervals smaller or the time intervals smaller, you would get the average speed for a smaller duration. If you could imagine making the interval very small your calculation would approach the speed at an instant. This may be difficult to do, But we could definitely imagine doing it. What we are trying to do here is to use a definition valid for an interval of time to find the speed at an instant, which is zero duration. The right kind of mathematics required to do that would lead us to an approximation to the value.
Do remember that the instantaneous speed is physically very meaningful. After all a moving body is moving at all moments of time when it is moving and must have a certian value for its speed. In fact the speed worked out over a duration would be same as the speed at any instant if the car did not speed up or slow down during the motion. So we have no quarrel with the physics of it. The problem is with the right kind of math needed for that. We say here that if the speed is worked for a sufficiently small interval it would be a good approximation to instantaneous speed.
Now imagine you note the odometer readings at different times. You can then plot the points on a graph. You draw a smooth curve between all those points. You can work out the speed by reading off the positions of the car at two different instants of time. (Mind you we plotted the graph by noting the distance at specific times, but once having drawn the graph you could read the positions at any time. Even at times when you had no readings for the distance. The interpolation between the points allows you do do that.
Suppose you wish to find the speed at t = 3 s. You could find the positions at t =2 s and t = 4 s from the curve. Distance for the duration of 2 s is the difference in the postions. And you can find the average speed by distance/ time. this ratio is the slope of the line joining the two points on the curve at t =2 s and t = 4 s. You can approach t=3 s by noting positions at t=2.1s and t=3.9 s. Now the duration is 1.8 s and the slope of the line is the average speed for this duration. Decreasing the interval furthur allows you to approach the instant t = 3 s. And at t = 3 s, though you do not have two points on the curve anymore, the slope is that of the tangent to the curve at t = 3 s.
Applet shows a graph of distance versus time of a body moving with a varying speed. To find the speed you would need positions of the body at two different instants of time. You can get these positions from the graph. The difference between the co-ordinates (which can be read off the y-axis) at the two instants will give us the distance. speed then would be delta_y / delta_x. ( Here delta_y on the graph stands for distance and delta_x on the graph for the time interval ). This calculation then is the delta_x/delta_t for the body and this would be average speed for the duration. In the applet delta_x is shown as orange line and delta_t as the pink line. Their ratio is the slope of the chord shown in blue. If you reduce the time interval the distance would decrease and you will get the ratios of the sides of a smaller triangle. This ratio is the slope of the blue line. As the duration becomes smaller, the blue line becomes smaller and approaches the tangent to the curve. The length of the blue line itself should not matter here. It is the slope of the line we are interested in. If necessary you could extend the blue line and measure its slope. When the length of the blue line is zero, we will draw a tangent to the curve and that would give us the required instantaneous speed. We now no longer have the values for distance and time interval, but we know their ratio to be slope of the tangent to the curve.
There are just two parameters that can be changed in the applet. The time at which the instantaneous speed is to be determined is set by using the upper scrollbar and the duration over which the average speed is caluculated is set by the lower scroll bar. At each of the values of upper scrollbar change the value of the duration through the entire range and see what happens.
It is also interesting to set the value of the lower scroll bar to zero, this would give the instantaneous speed and then vary the values of the upper scroll-bar through its full range.