Thread: relative velocities

A 136-m-wide river flows due east at a uniform speed of 9 m/s. A boat with a speed of 1 m/s relative to the water leaves the south bank pointed in a direction 16° west of north. How long does the boat take to cross the river? Assume an xy coordinate system with the positive direction of the x axis due east and the positive direction of the y axis due north.

Thanks in advance.

Originally Posted by winterwyrm

A 136-m-wide river flows due east at a uniform speed of 9 m/s. A boat with a speed of 1 m/s relative to the water leaves the south bank pointed in a direction 16° west of north. How long does the boat take to cross the river? Assume an xy coordinate system with the positive direction of the x axis due east and the positive direction of the y axis due north.

Thanks in advance.

The current is due East.
The boat is due N 16deg W.
So the boat is going against the current in some way.

The boat's speed is 1 m/sec relative to the speed of the current. In order for the boat to move to its direction, the due-west component of the boat's speed must be 1 m/sec relative to the current.
So if the boat were to travel in a straight line, the relative velocity, V, of the boat along that straight line is
V*cos(90 -16 deg) = 1
V = 1 / cos(74 deg) = 3.62796 m/sec, due N 16deg W.

The straight crossing line, d, is
d*sin(74deg) = 136
d = 136 / sin(74deg) = 141.48072 meters

d is also,
d = 3.62796*t
where t is in seconds

So,
3.62796(t) = 141.48072
t = 141.48072 / 3.62796
t = 39 seconds ------the time the boat will cross the river.