A small boat is headed for a harbor 30 km directly northwest of its current posi

Question

A small boat is headed for a harbor 30 km directly northwest of its current position whenit is suddenly engulfed in heavy fog. The captain maintains acompass bearing of northwest and a speed of 9.4 km/h relative to the water. The fog lifts 3.0 hlater and the captain notes that he is now exactly 4.2 km south of the harbor. (a) What was the average speed of the currentduring those 3.0 h?
km/h

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

(c) What would its travel time have been if it had followed astraight course?
h A small boat is headed for a harbor 30 km directly northwest of its current position whenit is suddenly engulfed in heavy fog. The captain maintains acompass bearing of northwest and a speed of 9.4 km/h relative to the water. The fog lifts 3.0 hlater and the captain notes that he is now exactly 4.2 km south of the harbor. (a) What was the average speed of the currentduring those 3.0 h?
km/h

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

(c) What would its travel time have been if it had followed astraight course?
h A small boat is headed for a harbor 30 km directly northwest of its current position whenit is suddenly engulfed in heavy fog. The captain maintains acompass bearing of northwest and a speed of 9.4 km/h relative to the water. The fog lifts 3.0 hlater and the captain notes that he is now exactly 4.2 km south of the harbor. (a) What was the average speed of the currentduring those 3.0 h?
km/h

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

(c) What would its travel time have been if it had followed astraight course?
h (a) What was the average speed of the currentduring those 3.0 h?
km/h

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

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

Explanation / Answer

  in the given problem let us assume that theorigin be at the position of the boat when it was engulfed by thefog    for simplicity let us take that the x and ydirections to be east and north respectively    vbw be the velocity of the boatrelative to the water    vbs be the velocity of the boatrelative to the shore    vws be the velocity ofthe water relative to the shore    then we can write that    vbs = vbw +vws     is the angle of vbs withrespect to x (east) direction     is the angle of vbs withrespect to x (east) direction (a)    first we find the position vector for theboat at t = 3 h    rboat = [(30 km)(cos135o) t] i^ + [(30 km) (sin135o) t - 4.2km] j^    the coordinates of the boat at t = 3 h    rx = [(9.4 km / h)cos135o + vws cos] (3 h)and    ry = [(9.4 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    now the velocity will be    vws = vx /cos

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