Racing cars driven by Chris and Kelly are side by side at...

Key Concepts

Midpoint Rule

The Midpoint Rule is a numerical integration technique used to approximate the definite integral of a function. It does so by dividing the interval into subintervals, then using the value of the function at the midpoint of each subinterval to form a rectangle whose area approximates the area under the curve. This method is particularly useful when you have values of a function at discrete points, such as from a data table.

Velocity and Displacement Relationship

In the context of motion, the displacement of an object is obtained by integrating its velocity over time. This relationship is fundamental in kinematics, as it connects the instantaneous rate of change (velocity) with the accumulated change (displacement). When the velocity data is provided as discrete values, numerical integration methods like the Midpoint Rule can be used to estimate the total distance traveled.

Numerical Integration from Data Tables

Numerical integration from data tables involves approximating the value of an integral based on discrete measurements rather than a continuous function. Techniques such as the Midpoint Rule allow for the estimation of cumulative quantities, like distance in this case, by summing the approximate areas (or contributions) from individual time intervals. This is especially useful in practical scenarios where only limited data points are available.

Transcript

00:01 Racing cars driven by chris and kelly are side by side at the start of a race.

00:06 The table shows the velocities of each car in miles per hour during the first 10 seconds of the race.

00:12 Use the midpoint rule to estimate how much further kelly travels than chris does during the first 10 seconds.

00:19 Okay, to use the midpoint rule, we're going to want to break up our time intervals so that we can actually use data at the midpoint.

00:28 So what do i mean by that? if we break up our time intervals into two second intervals, so for example, 0 to 2, 2 to 4, 4 to 6, 6 to 8, and 6 to 10, this allows us to use midpoints, because in the first interval from 0 to 2, we have a midpoint at t equals 1.

00:48 In the next interval, we have a midpoint of t equals 3 and so on and so forth.

00:53 We have 5, 7, and 9.

00:56 So we're going to be using data at the midpoints.

01:00 Apply the midpoint rule.

01:03 So the idea here is that we want to figure out how much farther kelly travels than chris does in each of these, each of these intervals, and then we're just going to add them up at the end.

01:15 So to start, let's look at the first interval.

01:17 0 less than or equal to t less than or equal to 2.

01:22 So here we want to figure out how much farther does kelly travel than chris in this interval.

01:28 So how do we figure out distance, the total, the difference in their position, that's going to to be velocity times time.

01:37 So let's call this d1, like the difference in their distances are in the first interval.

01:44 So that's going to be velocity times time.

01:47 But since we're looking at the difference in their positions, we're going to look at the difference in their velocities.

01:52 So that's going to be vk minus vc, but we're looking at the midpoint, which is time t equals 1.

01:59 So, okay, let's write it out.

02:02 Vk minus vc.

02:04 Then we're going to multiply by delta t.

02:08 That's going to give us the total difference in their distance for this time interval.

02:14 So we use the midpoint at time t equals 1.

02:17 Let's write that out.

02:19 T equals 1.

02:21 T equals 1.

02:21 Good.

02:22 And what we're going to do is we're going to use the values from t equals 1.

02:27 So at t equals 1, vk is 22 minus vc is 20.

02:35 Notice these are in miles per hour.

02:38 Our delta t here is to seconds.

02:42 So we're actually going to want to convert our seconds to hours.

02:47 So how do we convert seconds to hours? well, there's 60 seconds in one minute.

02:57 Oh, we want to write it in the opposite way so that our units cancel.

03:03 Okay, there's 60 seconds in one minute and 60 minutes in one hour.

03:08 So 60 seconds in one minute and 60 minutes in one hour.

03:20 Since this is miles per hour, we see here seconds and seconds cancel, minutes and minutes cancel.

03:26 And then this is miles per hour.

03:29 Hours cancels with hours...

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