Experiment suggests that a falling body will fall...

Experiment suggests that a falling body will fall approximately $16 t^{2}$ feet in $t$ seconds.
(a) How far will it fall between $t=0$ and $t=1 ?$
(b) How far will it fall between $t=1$ and $t=2 ?$
(c) What is its average velocity on the interval $2 \leq t \leq 3 ?$

Key Concepts

Displacement Function

The displacement function is a mathematical model that expresses the position or distance traveled by an object as a function of time. It is used to represent how far an object has fallen or moved from its starting point within a specific time period.

Interval Analysis

Interval analysis involves calculating the change in a function’s value over a specific range of the independent variable (in this case, time). This concept is crucial for determining the displacement between two time instants by evaluating the difference in the function’s outputs at those endpoints.

Average Velocity

Average velocity is defined as the total displacement divided by the total time taken over a given interval. It provides an overall measure of how quickly an object moves between two distinct time points, which is essential in understanding motion in kinematics.

Transcript

00:01 So an essential idea here is to think about average velocity.

00:05 The average velocity is the change in height or change in position generally over the change in time.

00:12 And this might remind you of slope, right? it's like rise over run.

00:16 We're just using quantities from physics, namely height and time.

00:21 So let's make a little table that tells us the time and the height.

00:27 So at zero seconds, we have 16 times zero squared.

00:30 That's zero.

00:32 At one, we have 16 times one squared, which is 16.

00:36 We can plug in two as well.

00:39 Let's plug in three for a good measure.

00:43 So how far does this object fall between zero and one seconds? that's a change in 16 feet.

00:52 So that's the answer to part a.

00:55 How about between one and two seconds? well, 64 minus 16 is 48 feet.

01:01 And that's the answer to part b.

01:03 Part c, is interested in us analyzing the average velocity between times two and three.

01:12 Well, the change in height is 144 minus 64 or 80.

01:18 Now we need to divide by the change in time as well, but that's us three minus two.

01:25 So our average velocity is 80 feet in one second or 80 feet per second.

01:31 And note that's only the average velocity starting at time two and ending at time three.

01:35 That interval.

01:37 How about the average velocity for the interval 3 to 3 .01? well, we could plug 3 .01 in, 16 times 3 .01 squared is 144 .96.

01:56 So let's compute our average velocity again, using our formula, the change in height or position would be those two values subtracted...

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