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Math Help - vectors problem

  1. #1
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    vectors problem

    1)Adam can swim at the rate of 2 km/h in still water.At what angle to the bank of a river must he head if he wants to swim directly across the river and the current in the river moves at the rate of 1 km/h?
    2) A boat heads 15degrees west of of north with a water speed of 12 m/s. Determine its resultant velocity , relative to the ground, when it encounters a 5 m/s current from 15 degrees north of east.

    Thank you in advance!
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  2. #2
    Super Member redsoxfan325's Avatar
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    Quote Originally Posted by anna12345 View Post
    1)Adam can swim at the rate of 2 km/h in still water.At what angle to the bank of a river must he head if he wants to swim directly across the river and the current in the river moves at the rate of 1 km/h?
    2) A boat heads 15degrees west of of north with a water speed of 12 m/s. Determine its resultant velocity , relative to the ground, when it encounters a 5 m/s current from 15 degrees north of east.

    Thank you in advance!
    1.) We can set up a triangle. His velocity of 2 km/h is the hypotenuse of the triangle and the current's velocity of 1 km/h is the opposite height. Thus the angle we're looking for is \sin^{-1}(1/2) = \boxed{30^o}

    2.) The 15 degrees is the direction of the vectors and the 12 m/s and 5 m/s are the magnitudes of the vectors. If you draw out the vectors, you will see that they intersect at a right angle. Thus, using the "tip-to-tail" method of adding vectors is just the Pythagorean Theorem: \sqrt{12^2+5^2}=13. So it is going 13 m/s. To find its angle with the origin, you need to start by taking \tan^{-1}(5/12)\approx 22.62^o. However, this gives you the one of the angles of the 5-12-13 triangle, not the angle the 13 m/s vector makes with the origin. To find this angle, subtract 15^o from the angle you found above to get 7.62^o. So the boat travels at 13 m/s in the direction 7.62^o east of north.

    I'm sure that was confusing but if you draw out the picture it should be easier to understand.
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