We suppose that a ship, that is at the position $(1, 0)$ of a nautical map (with the North at the positive direction $y$) and it "sees" a rock at the position $(2, 4)$, is directed to North and is travelling $4$ knots in the relation to the water. There is a current of 1 knot that is directed to the east side. The units on the map are nautical miles, $1$ knot=$1$ nautical mile per hour.
a) If there weren't the current, which vector $\overrightarrow{u}$ would represent the velocity of the ship in relation to the sea floor?
b) If the ship was just following the current, which vector $\overrightarrow{v}$ would represent the velocity in relation to the sea floor?
c) Which vector $\overrightarrow{w}$ represents the total velocity of the ship?
d) Where will the ship be after $1$ hour?
e) Does the captain have to change direction?
f) What would happen if the rock was an iceberg?
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What does it mean "...which vector would represent the velocity of the ship in relation to the see floor?" ??
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In my book there is the following:
If a body moves with uniform velocity on a line, then the velocity vector is a displacement vector from its position at a moment till the position $1$ unit time later.
I have done the following:
a) The ship is travelling to North 4 knots and there is no current. That means that we have the following:
So $\overrightarrow{u}=(1, 4)-(1, 0)=(0, 4)$.
Is it correct??
b) The ship follows the current.
In my book at an other example there is the following:
Could you explain to me why it is like that??
c) How can we find the total velocity of the ship??
d) The position of the ship in $1$ hour is "$\text{ Initial position + Total velocity} =(1, 0)+\overrightarrow{w}$, right??
e) After $1$ hour the ship will be at the position $(4, 8)$ which is the same line as the position of the rock $(2, 4)$. So, the captain has to change direction.
Is this correct??
f) What would be in that case??
(a) It is correct.
(b) If "the ship follows the current" means it does not have its own velocity but just being moved by the current, then you should look at the current velocity, which is 1 knot to the right. So the vector is $(1,0)$.
(c) The picture from your book actually explained this situation. When the ship has its own velocity to the north, and is also affected by the current to the right, the total effect is like in the picture, it is moved 4 upward and 1 to the right, so $(1,4)$.
(d) Correct.
(e) After 1 hour it should be at exactly $(2,4)$.
(f) I assume the iceberg would move with the current. In that case, the captain does not need to change direction.