A boat with a broken motor can travel at only one speed: 45 m/s while on still w

Question

A boat with a broken motor can travel at only one speed: 45 m/s while on still water. This boat is placed in a 1200 m wide river that flows from west to east with a current of 15 m/s. Assuming that the boat is facing directly north, and starts on the southern shore of the river find the following: 1. a) What is the velocity of the boat as seen from a helicopter hovering directly above the river? b) How much time is required for the boat to reach the northern shore of the river? c) How far down stream will the boat end up? d) What is the final displacement of the boat? At what angle relative to the shore would the boat have to face in order to end up directly across the river? e) The same boat from the previous problem is placed in the same river. The only difference is that now the boat starts off facing 20° north of west. Answer parts a-e from question 1 based on this new information.

Explanation / Answer

Taking unit vector along east as i and unit vector along north as j.
Velocity of Boat wrt river = 45 j
Velocity of river = 15 i
a) Velocity of boat = Velocity of boat wrt river + velocity of river = 15i + 45j
Magnitude of velocity = (152 + 452) 1/2 = 47.4 m/s
As boat crosses the river, it's dispacement = xi + 1200 j
b) Time taken to cross river = displacement along north / velocity along north = 1200/45 = 26.7 sec

c)displacement along the stream, is displacement along east = velocity along east x time = 12x26.7 = 400 m

d) final displacement 400i + 1200j
Magnitude = (4002 + 12002)1/2 = 1265 m

e) For final velocity along j direction , component of velocity of boat wrt river should be equal and opposite to the velocity of river, that is -15i
so angle relative to shore, theta = cos-1(15/45) = 70.5 deg

For second part take boat velocity wrt river = 45( -cos(20)i + sin20j)
This gives velocity of boat = Vbr + Vr
using htis we can answer all parts a to d as above.

  

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