A river that is flowing due west is crossed by a boat travelling from the south bank to the north bank. The boat can travel at 30 m/s in still water and the river is moving at 10 m/s.
Determine the resultant velocity for the boat when it is travelling directly across the river. Please determine the magnitude only.
To determine the resultant velocity of the boat when it is traveling directly across the river, we can use vector addition.
Let the velocity of the boat relative to the water be Vb, the velocity of the river be Vr, and the resultant velocity be Vr.
The velocity of the boat relative to the water is 30 m/s in the west direction. The velocity of the river is 10 m/s in the west direction. The direction in which the boat travels from the south bank to the north bank is north.
Using vector addition, we have:
Vr = Vb + Vr
Since the boat is traveling directly across the river, the angle between the boat's velocity and the river's velocity is 90 degrees.
Using the Pythagorean theorem, we can find the magnitude of the resultant velocity:
|Vr| = sqrt( (30 m/s)^2 + (10 m/s)^2 )
|Vr| = sqrt( 900 + 100 )
|Vr| = sqrt( 1000 )
|Vr| ≈ 31.62 m/s
Therefore, the magnitude of the resultant velocity for the boat when it is traveling directly across the river is approximately 31.62 m/s.