Two cars, A and B, start side by side and accelerate from...

Name: Problem 47, Chapter 5: Calculus
Uploaded: 2021-03-26T17:40:45-08:00
Duration: 10 min 42 s
Channel: Chris Trentman
Description: Two cars, A and B, start side by side and accelerate from rest. The figure shows the graphs of their velocity functions. (a) Which car is ahead after one minute? Explain. (b) What is the meaning of the area of the shaded region? (c) Which car is ahead after two minutes? Explain. (d) Estimate the time at which the cars are again side by side.

Two cars, A and B, start side by side and accelerate from rest. The figure shows the graphs of their velocity functions. (a) Which car is ahead after one minute? Explain. (b) What is the meaning of the area of the shaded region? (c) Which car is ahead after two minutes? Explain. (d) Estimate the time at which the cars are again side by side.

Two cars, A and B, start side by side and accelerate from rest. The figure shows the graphs of their velocity functions.
(a) Which car is ahead after one minute? Explain.
(b) What is the meaning of the area of the shaded region?
(c) Which car is ahead after two minutes? Explain.
(d) Estimate the time at which the cars are again side by side.

Key Concepts

Velocity-Time Graphs

Velocity-time graphs are graphical representations that show how an object’s velocity changes over time. The slope of the graph at any point indicates the object's acceleration, and the overall shape of the curve provides insights into the object's motion profile, such as whether the velocity is increasing, decreasing, or constant.

Displacement via Integration

The area under a velocity-time graph represents the displacement or distance traveled by the object over a specific time interval. By calculating the definite integral of the velocity function, one obtains the cumulative distance covered, which is essential for comparing motions over time.

Relative Motion

Relative motion involves comparing the displacement or position of two or more objects as they move. By comparing the areas under their respective velocity-time curves (i.e., their displacements), one can determine which object is ahead or if they are level with each other at particular moments.

Intersection of Displacement Curves

When two objects have travelled the same overall distance, they are side by side. This concept is represented graphically by the intersection of their cumulative displacement (or position) functions, indicating the moments when the objects have equal displacement despite potentially different velocity profiles.

Transcript

00:03 We are given a question about two cars and we're asked to answer this question using areas between curves.

00:12 So we're told that two cars a and b start side by side and accelerate from rest.

00:18 And the figure in exercise 53 shows the graphs of each of their velocity functions.

00:25 In part a, we're asked which car is ahead after one minute and to explain.

00:34 To determine this, what we want to determine which car is ahead? a farther distance than the other.

00:44 So they both start side by side.

00:48 Recall that, since we're given the velocities of the car, we know that the displacement, which one could call s of t, this is, in fact, the integral of v of t, dt, and so it follows that we see in this figure, the velocity for car a is greater than or equal to zero, as is the velocity for b.

01:49 And so it follows that the distance traveled by car a at time t.

01:57 This is simply the integral from time zero to time t of the velocity of car a.

02:08 And i guess i'll make this u, d, u.

02:13 Distance traveled by car b by time t is equal to the integral from 0 to t of v.

02:23 Sub b of u, du.

02:30 So we want to see which has the greater distance after one minute.

02:38 So looking at our graph, da of 1, this is the integral from 0 to 1 of v .a.

02:52 U .d .u.

02:53 So the area under a, which is clearly strictly greater than the integral from 0 to 1, of b .b of udu, the area under the velocity curve, or b, which is equal to db of 1.

03:13 The distance traveled by b after one minute, and therefore the answer to part a is car a.

03:26 Okay, then in part b, we're asked what is the meaning of the area of the shaded region? well, the area of the shaded region, this is in fact the integral from 0 to 1 of the velocity of a minus the velocity of b, d .u.

04:00 Which, of course, this is equal to the integral from 0 to 1 of the velocity of a, u, du, minus the integral from 0 .1.

04:08 To 1 of the velocity of b, udu, by addativity.

04:17 And of course, this is equal to the distance traveled by car a after one minute, minus the distance traveled by car b after one minute...

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