A small boat is headed for a harbor 35 km directly northwest of its current posi

Question

A small boat is headed for a harbor 35 km directly northwest of its current position when it is suddenly engulfed in heavy fog. The captain maintains a compass bearing of northwest and a speed of 10.1 km/h relative to the water. The fog lifts 3.0 h later and the captain notes that he is now exactly 3.6 km south of the harbor.

(a) What was the average speed of the current during those 3.0 h?
km/h

(b) In what direction should the boat have been heading to reach its destination along a straight course?
°west of north

(c) What would its travel time have been if it had followed a straight course?
h

Explanation / Answer

in the given problem let us assume that the origin be at the position of the boat when it was engulfed by the fog for simplicity let us take that the x and y directions to be east and north respectively vbw be the velocity of the boat relative to the water vbs be the velocity of the boat relative to the shore vws be the velocity of the water relative to the shore then we can write that vbs = vbw +vws ? is the angle of vbs with respect to x (east) direction (a) first we find the position vector for theboat at t = 3 h rboat = [(35 km)(cos135o) t] i^ + [(35 km) (sin135o) t - 3.6km] j^ the coordinates of the boat at t = 3 h rx = [(10.1 km / h)cos135o + vws cos?] (3 h)and ry = [(10.1 km / h)sin135o + vws sin?] (3 h) simplify to get x vwscos?= - ....... km / h x vwssin?= - ....... km / h dividing them we get tan? = ............ ? = tan-1() = ......o now the velocity will be vws = vx /cos?

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