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In this question-

A motorboat is racing towards north at 25km/h and the water current in that region is 10km/hr in the direction of 60 degree east of south. Find the resultant velocity of the boat.

The first part is quite easy and we get 21.8 approx as the magnitude of the resultant.

My doubt is in the 2nd part of the question. How do we calculate the direction using the normal method. one way is to use the sine formula and say that R/sin(a)=velocity of current/sin(alpha) where a is angle between the vectors and alpha is the angle of resultant R with the north direction. this gives us the angle with the north that is 23.4 degrees which is correct but how does one use the normal tan(alpha) method which works in questions where alpha is less that 90 or equal to 90.

By the tan(alpha) method i am referring to this formula- tan(alpha)=Asin(a)/(b+A*cos(a)) one of the book says that it is tan(alpha)=10sin120/(25+10cos120) which gives the correct answer also but i cant understand how we get this. after making a parallelogram if the angle b/w vectors is less than 90 or 90 we just extend one of the sides and get 2 right angle triangles. in this case we get only one.please explain how do we do this.

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  • $\begingroup$ How can you get the length of the vector without knowing the direction? And does "towards north" mean that it is heading (i.e. pointing) North, or that its velocity w.r.t. the earth is North (i.e. that it is the resultant that is pointing North). "At 25 km/h" - is that with respect to the water, or the land? It is decidedly unclear. Perhaps if you draw a diagram (ALWAYS draw a diagram) we can see what you mean. And when you have done so, the answer will become quite obvious, I think... Just compute the resultant velocity to the East and North separately. $\endgroup$ – Floris Sep 12 '14 at 19:53
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This is the diagram you need to draw:

enter image description here

The red arrow is the net velocity of the boat - the sum of 25 km/h going due North, and a current of 10 m/s at 60 degrees East of South. (not drawn to scale)

Just do simple math to determine the size of the two green segments - then compute the angle (heading) from the arctan of their ratio.

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