What is the difference between average speed and instantaneous speed?

Average speed is the total distance travelled divided by the total time taken.

Instantaneous speed is the speed at a particular moment in time, usually measured using calculus. It is the limit of the average speed as the time interval approaches zero. In other words, it is the speed at a specific point in time, rather than an average over a period of time.

To calculate average speed, we use the formula:

Average speed = total distance travelled ÷ total time taken

For example, if a car travels 100 miles in 2 hours, its average speed is:

Average speed = 100 miles ÷ 2 hours = 50 miles per hour

Instantaneous speed, on the other hand, is calculated using calculus. We can find the instantaneous speed of an object at a particular moment in time by finding the derivative of its position function with respect to time. In other words, we find the slope of the tangent line to the position function at that point.

For example, if an object's position function is given by:

s(t) = 2t^2 + 3t + 1

We can find its instantaneous speed at time t = 2 by finding the derivative of the position function:

s'(t) = 4t + 3

Then we substitute t = 2 into the derivative to get the instantaneous speed:

s'(2) = 4(2) + 3 = 11

So the object's instantaneous speed at time t = 2 is 11 units per time (e.g. meters per second, miles per hour, etc.).