A river flows due east at 1.50 m / s. A boat crosses the river from the south shore to the nort

A river flows due east at 1.50 m / s. A boat crosses the...

Key Concepts

Uniform Motion and Displacement

Uniform motion refers to movement at a constant speed in a given direction, leading to straightforward calculations of displacement as the product of speed and time. This principle is fundamental when determining how far an object has traveled in a specific direction over a period of time.

Component Analysis

Component analysis involves breaking down vectors into their perpendicular components, typically along axes such as horizontal and vertical. This approach simplifies the process of vector addition and is crucial for analyzing how different directional movements contribute to the net displacement or velocity.

Vector Addition

Vector addition is the process of combining two or more vectors, taking into account both their magnitudes and directions, to produce a single resultant vector. This technique is essential for determining the overall velocity when multiple influences, such as a boat's propulsion and a river's current, act simultaneously.

Relative Velocity

Relative velocity is a concept used to determine how the velocity of an object is perceived from a particular frame of reference. It involves understanding the object’s velocity in one medium and then adding the velocity of that medium relative to another frame, such as the shore or ground.

Transcript

00:01 So in this problem, we're going to use, again, the concepts of relative velocity, as well as vector addition, and the components of these vectors in order to figure out the velocity of this boat relative to the earth.

00:14 And then we're also going to apply this to figure out how much distance the boat has traveled in either direction as it crosses the river.

00:22 So here we have a river which is shown in blue, and we have the water flowing directly to the right, which is labeled as east.

00:32 So this way is east along the x -axis, such that upwards is north.

00:38 And we're told that a boat travels along the river, and it always moves directly north relative to the water.

00:47 So while the boat is in the water, the velocity of the boat relative to the water always points directly upwards.

00:53 North, which is shown here.

00:56 The velocity of the boat relative to the earth will then be the addition of these two velocities, since the velocity of the boat relative to the water is upwards, and the velocity is moving to the right.

01:09 The people observing from the outside, see someone down here, will see the boat moving with the water.

01:19 So you have to add the velocity of the boat and the velocity of the water to find the velocity of the boat relative to the earth, which is what we see.

01:26 What is what we see here? the other way you can verify this is that from the equations in the book, we know that the velocity of the boat relative to the water with the velocity of the boat relative to the earth minus the velocity of the water relative to the earth.

01:44 And if we rearrange this, the velocity of the boat relative to the earth, we have the velocity of the boat relative to the water, plus the velocity of the boat plus the velocity of the water relative to the earth water relative to the earth as so and we're given or we define our axes such as the y axis points north and the x axis points east and since each of these point along the exact direction we can then just substitute these of representation for these vectors in so this would be 10 y hat plus 1 .5 x hat since we are given that the boat moves 10 meters second north relative to the water and the water moves 1 .5 meters per second east and so we want to know the magnitude of this velocity not what its components are want to know how fast the boat is moving and so that will be the sum of the squares of the components so 10 squared plus 1 .5 squared which is around 10 .11 meters per second we can also verify that this has to be true because the boat will be moving in two separate since the boat is moving north relative to the water, and the water is moving to the moving east.

03:04 So therefore, the boat has to be moving in two separate directions relative to somebody watching on the outside, which means that it has to have a velocity greater than its velocity relative to the water, since it's going to be moving in two separate directions.

03:20 It can't be moving slower if it's moving in two separate directions, because that wouldn't make any sense.

03:29 So from here, we know how fast the boat is moving.

03:34 So then if we're told that the distance between the shores, which are shown here, is 300 meters, and we know that the boat moves to the up and to the right, then we want to know how far, say, call this delta x, the boat moves to the right after it reaches the shore.

03:54 So say it starts on one shore, leaves with the constant...

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