A river flows due east at 1.91m/s.A boat crosses the river from the south shore to the north shore by maintaining a constant velocity of 10.3m/s due north relative to the water.
A.What is the velocity of the boat relative to shore?
How far downstream has the boat moved by the time it reaches the North shore?
To find the velocity of the boat relative to shore, we need to use vector addition.
First, we break down the velocity of the river and the velocity of the boat into their respective components.
The river is flowing due east at a velocity of 1.91 m/s. Since the boat is crossing the river from the south shore to the north shore, the component of the river's velocity that affects the boat's motion is northward. Therefore, the river's velocity component is 0 m/s in the east direction and 1.91 m/s in the north direction.
The boat is moving with a constant velocity of 10.3 m/s due north relative to the water. Therefore, the boat's velocity component is 0 m/s in the east direction and 10.3 m/s in the north direction.
To find the velocity of the boat relative to the shore, we add the respective components together.
So, the velocity of the boat relative to the shore is 0 m/s in the east direction and (10.3 m/s + 1.91 m/s) = 12.21 m/s in the north direction.
Therefore, the velocity of the boat relative to the shore is 12.21 m/s due north.
To find how far downstream the boat has moved by the time it reaches the North shore, we need to calculate the displacement caused by the river's motion during the time it took the boat to cross the river.
The time it takes for the boat to cross the river is calculated by dividing the width of the river by the boat's velocity component in the north direction.
Let's assume the width of the river is represented by the symbol "w".
Therefore, the time taken to cross the river is w / (10.3 m/s).
During this time, the boat is also moving due east at a velocity of 1.91 m/s, so the distance downstream is given by (1.91 m/s) * (w / (10.3 m/s)).
Therefore, the distance downstream is (1.91/10.3) * w.
You can now substitute the value of "w" to find the exact distance downstream the boat has moved by the time it reaches the North shore.