# resolving vectors

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I really dont know where to go with this one.

A boat sails across a straight river of uniform width W, starting from a point O on one bank of the river. The velocity of the river at a distance y from the bank is u(y)=ay(W-y), where a is a positive constant. The boat travels at a constant speed v relative to the current and steers a course set at a constant angle p between 0 and pi. in the downstream direction.

a) show that the velocity of the boat is

(u+vcosP)e1+(vsinP)e2.

b)at what time does the boat reach the other bank?

c) show that when the boat has reached the other bank, the downstream distance it has travelled is equatl to

$$\frac{aW^3}{6vsinP}+WcotP$$

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There are a few basic rules to follow when doing this kinda question.

1) Firstly the velocity of the boat squared will (using Pythagoras' Theorem) the horizontal component squared add the vertical component squared.

e.g. If it's moving 10m/s towards the other bank and 5m/s down the river then it's overall velocity is:

root (10^2 + 5^2)

root (100 + 25)

root (125)

11.18033989

2) Remember the horizontal component of a velocity does not effect the vertical component.

e.g. So for this question the fact that there is a current in the water will not change the time it takes for the boat to cross the river. If it would normally take 10sec and then you add a current of x the time taken to cross the river will still be 10sec.

3) Once it has travelled a known distance (width of the river) you should know the time taken. Knowing the downstream velocity and the time it is sailing for should let you know the total distance downstream.

Does that help?

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