A river flowing with a velocity of 2m/s due south and a boat wants to move with a speed of 10m/s across the river. At what speed and direction should the boat move
In order to move across the river with a speed of 10m/s, the boat needs to travel in a direction that is a combination of both the velocity of the river and the desired speed across the river. This can be determined using vector addition.
Let's break down the velocity of the river into its x and y components:
Vriver_x = 0 m/s
Vriver_y = -2 m/s
Now, let's calculate the components of the boat's desired speed across the river:
Vboat_x = 10 m/s
Vboat_y = 0 m/s
To determine the speed and direction of the boat, we need to add the x and y components of the boat's speed with the x and y components of the river's velocity:
Vresult_x = Vboat_x + Vriver_x = 10 m/s + 0 m/s = 10 m/s
Vresult_y = Vboat_y + Vriver_y = 0 m/s + (-2 m/s) = -2 m/s
The magnitude of the resultant velocity is given by:
Vresult = sqrt(Vresult_x^2 + Vresult_y^2) = sqrt(10^2 + (-2)^2) = sqrt(104) ≈ 10.2 m/s
Therefore, the boat should move with a speed of approximately 10.2 m/s at an angle of arctan(-2/10) = -11.3 degrees or approximately 11.3 degrees west of south.