A motor boat is racing towards north at 25 m/s and the water current in that region is 10 m/s in the direction of 60 degree east of south. Find the resultant velocity of boat

Asked by Anil | 28th May, 2017, 06:58: PM

Expert Answer:

 
begin mathsize 12px style Speed space of space the space boat space is space straight v subscript straight b equals 25 space km divided by straight h space northwards
Speed space of space the space water space current space is space straight v subscript straight w equals 10 space km divided by straight h space 60 degree space east space of space south.

This space is space shown space in space the space figure space above.
From space the space geometry space of space the space figure comma space the space angle space between space the space boat space and space the space water space current space is space 120 degree

Hence comma space the space resultant space speed space of space the space boat space is
straight v subscript br equals square root of straight v subscript straight b superscript 2 plus straight v subscript straight w superscript 2 plus 2 straight v subscript straight b straight v subscript straight w cos 120 end root
therefore straight v subscript br equals square root of 25 squared plus 10 squared plus 2 cross times 25 cross times 10 cross times open parentheses negative 0.5 close parentheses end root
therefore straight v subscript br equals square root of 625 plus 100 minus 250 end root equals square root of 475 equals 21.8 space km divided by straight h
Now comma space in space order space to space find space the space direction space let space us space suppose space that space the space resultant space is space making space an space angle space straight alpha space with space the space boat
Using space tangent space rule comma
tanα equals fraction numerator Bsinθ over denominator straight A plus Bcosθ end fraction
here space straight A space is space the space speed space of space the space boat comma space straight B space is space the space speed space of space water space current space and space straight theta space is space the space angle space between space the space two.
therefore tanα equals fraction numerator straight v subscript straight w sin 120 over denominator straight v subscript straight b plus straight v subscript straight w cos 120 end fraction equals fraction numerator 10 sin 120 over denominator 25 plus 10 cos 120 end fraction equals fraction numerator 0.866 over denominator 25 minus 10 cross times 0.5 end fraction equals fraction numerator 8.66 over denominator 20 end fraction equals 0.433
therefore straight alpha equals 23.4 degree end style

Answered by Romal Bhansali | 29th May, 2017, 12:27: PM