Two canoeists in identical canoes exert the same effort...

Two canoeists in identical canoes exert the same effort paddling and hence maintain the same speed relaLive to the water, One paddles directly upstream (and moves upstream), whereas the other paddles directly downstream. With downstream as the positive direction. an observer on shore determines the velocities of the two canoes to be $-1.2 \mathrm{~m} / \mathrm{s}$ and $+2.9 \mathrm{~m} / \mathrm{s}$, respectively. (a) What is the speed of the water relative to the shore? (b) What is the speed of each canoe relative to the water?

Two canoeists in identical canoes exert the same effort paddling and hence maintain the same speed relaLive to the water, One paddles directly upstream (and moves upstream), whereas the other paddles directly downstream. With downstream as the positive direction. an observer on shore determines the velocities of the two canoes to be $-1.2 \mathrm{~m} / \mathrm{s}$ and $+2.9 \mathrm{~m} / \mathrm{s}$, respectively.
(a) What is the speed of the water relative to the shore?
(b) What is the speed of each canoe relative to the water?

Key Concepts

Sign Conventions

Sign conventions are rules that assign positive or negative values to vector quantities based on their directions. This concept is important when solving problems involving directional motion to ensure that upstream and downstream movements are correctly represented and combined using algebraic methods.

Vector Addition

Vector addition is a mathematical method used to combine velocities from different directions. In the context of motion in a river, the velocity of a canoe relative to the shore is found by vectorially adding the canoe's velocity relative to the water and the water's velocity relative to the shore, taking proper account of their directions.

Reference Frames

Reference frames are the perspectives from which measurements of physical quantities such as velocity are made. In problems involving moving water and canoes, for example, shifts between the frame of the water and the frame of the shore are crucial for establishing correct relationships between the speeds of the objects involved.

Relative Velocity

Relative velocity is the concept that describes the velocity of an object as observed from a particular frame of reference. It involves understanding that the observed speed of an object can differ depending on whether the observer is moving with or independent of the object, which is essential when analyzing problems that involve multiple moving mediums or platforms.

Transcript

00:01 In this problem, we're going to use the concept of relative velocity and apply it to a river in which we have two separate canoists paddling in two different directions with two different velocities relative to the earth.

00:14 And we need to figure out what the velocity of the water relative to the earth is.

00:20 And so we're given various information when we set the downstream to be to the right, such that the water is flowing to the right.

00:28 And we have that one canoeist is flowing upstream, which in this case would be to the left, with the velocity of 1 .2 meters per second.

00:36 And that's a magnitude.

00:37 Remember, the direction is in the negative x hat direction, such that the velocity vector of canoeer 1 with respect to the earth be negative 1 .2 x -hat moving in the negative x direction.

00:53 Similarly, we are given that the velocity of a second canoe are paddling downstream, so with the water is 2 .9 meters per second, such that its velocity vector is 2 .9 in the x -hat direction.

01:06 We're also given that both canoers are paddling with the same amount of effort, such that their velocities with respect to the water are the same.

01:15 And we have to equate their magnitudes here.

01:18 We can't equate the vectors because one canoer is moving left with respect to the water, and the other is moving to the right.

01:25 And so, well, if we take that into consideration, then we know that there'll be the vector of the same magnitude, but opposite direction.

01:34 So therefore, one is equal to the negative of the other.

01:40 So the velocity of canoe or one with respect to the water is equal to the negative vector of the velocity two with respect to the water.

01:49 And we need to figure out what the velocity of the water relative to the earth is.

01:53 And in order to do this, well, first let's write down what we know for the relative velocity of each of each canoeist.

02:01 We know that the velocity of canoeer 1 with respect to the water can be written as.

02:07 So we're writing the velocity with respect to two objects.

02:11 So we do the same pattern that we've seen from the book.

02:14 This will be the velocity of canoeer 1 with respect to the earth, minus the velocity of the water with respect to the earth.

02:21 We do the same thing with the velocity of canoe or two with respect to the water.

02:28 This will be the velocity of canoe or two with respect to the earth, minus the velocity of the water with respect to the earth.

02:36 Since the water is flowing the same for both canoists, these two are the same.

02:42 And since we're given, since we have this relationship down here, we can now insert these two expressions for the velocity of canoeer one and two with respect to the water...

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