Question

How do you find the average velocity over an interval?

Answer 1

Hint: Here we will use the concept of velocity and average velocity. Then we will assume a function whose value is within the interval. Then we will subtract the value of the function on two intervals and divide it by the difference between the intervals to get the average velocity.

Complete step-by-step answer:
The velocity of any object is the rate of change in the position of that object with respect to the time taken by it to do so.
Average velocity is defined as the change in position of an object divided by the time it had taken to do so.
Let \[f\left( x \right)\] be a function over an interval \[\left[ {a,b} \right]\].
Now to find the average velocity of the function over that interval we will subtract the function over the initial point from the function at the final point and divide the result by the difference of the interval. Therefore, we get
\[{v_{avg}} = \dfrac{{f\left( b \right) - f\left( a \right)}}{{b - a}}\]
Here, \[{v_{avg}} = \] Average velocity, \[f\left( b \right) = \] Value of function at point \[b\] and \[f\left( a \right) = \] Value of function at point \[a\].

Note:
Average velocity can also be used to calculate the amount of constant velocity that is required to give the same displacement every time it is applied on that object. The SI unit of velocity is (\[{\rm{m/s}}\]) as the distance unit is meter and time unit is second. Velocity is termed as a vector quantity because it has both magnitude and direction.
Formula for average velocity is given as:
\[{v_{avg}} = \dfrac{{\Delta x}}{{\Delta t}} = \dfrac{{{x_f} - {x_0}}}{{{t_f} - {t_0}}}\]
Where, \[{v_{avg}} = \] Average velocity, \[\Delta x = \] change in position, \[\Delta t = \] Time traveled, \[{x_f} = \] Final velocity, \[{x_0} = \] Initial Velocity, \[{t_f} = \] Final time taken\[{t_0} = \] Initial time.