(problem 1) The height of a vertically free-falling object at time seconds is given by
feet. Find the average velocity of the object from time seconds to time
seconds.
The positions at times and are The displacement is The average velocity of the
object over the time interval is
We compute average velocity to estimate instantaneous velocity.
Average Velocity
We will consider an object moving along a straight line, such as a vertically free-falling object. If the position of the object at time is denoted by , then the average velocity of the object from time to is given by In other words, average velocity is the change in position divided by the change in time. If the position is measured in feet and the time is measured in seconds, then average velocity will be measured in feet per second (ft/sec). The change in position, is also known as displacement.
example 1 The height of a falling object at time seconds is given by feet. Find the
average velocity of the object over the time interval .
We compute and the displacement, : The average velocity is Note that the negative sign in the average velocity indicates that the object is falling.
We compute and the displacement, : The average velocity is Note that the negative sign in the average velocity indicates that the object is falling.
Average velocity has a connection with the slope of a line. Recall that the slope of a line between two points, and is given by Thus, if we make a graph of position versus time, with time on the horizontal axis and position on the vertical axis, then the average velocity of a vertically free-falling object from time to time is the same as the slope of the line segment connecting the points and . See the figure below.
example 2 In the last example, with , we found that ft and ft. If we plot the points
and then the slope of the segment connecting them is the average velocity of the
falling object from time seconds to time seconds, as shown in the figure
below.
Instantaneous Velocity
To determine the velocity of an object at a particular moment in time, i.e., the instantaneous velocity, we find the average velocity over smaller and smaller time intervals.
example 3 The height of a falling object at time seconds is given by feet. Find the
average velocity of the object over the time intervals and .
First we will compute the height (in feet) at time and seconds:
First we will compute the height (in feet) at time and seconds:
The average velocities (in ft/sec) are:
We can now estimate the instantaneous velocity of the object at time seconds. Based on the average velocities over the smallest time intervals, namely, ft/sec and ft/sec, it seems that ft/sec would be a reasonable estimate for the instantaneous velocity of the falling object at time seconds. To improve our estimate, or have more confidence in it, we could compute the average velocity over even smaller time intervals.
(problem 3a) The height of a vertically free-falling object at time seconds is given by
Find the average velocity of the object over each of time intervals given below and
use your results to estimate the instantaneous velocity of the object at time,
seconds.
Instantaneous velocity at seconds: .
(problem 3b) The height of a vertical projectile at time seconds is given by Find the
average velocity of the object over each of time intervals given below and use
your results to estimate the instantaneous velocity of the object at time,
second.
Instantaneous velocity at second: .
The positive answer to this problem indicates that the projectile was rising at time
second.
The Tangent Line
We saw that the average velocity of a falling object can be represented by the slope of a line. The instantaneous velocity of a falling object can also be represented by the slope of a line. The graph below represents the position function for a falling object. The slope of the blue secant line represents the average velocity of the object over the time interval . The slope of the red tangent line represents the instantaneous velocity of the object at time .
In differential calculus, we study the tangent line and its applications.