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EDEXCEL M4 Relative Motion Help

Hi,

I am extremely stuck on Question 12 Exercise 1A in the Edexcel M4 textbook. The question is: A river flows at speed u. A boat is rowed with speed v relative to the river. The width of the river is w and the boat is to reach the opposite bank at a distance d downstream. Show that, if (uv/sqrt((w^2)+(d^2)))<v<u there are two directions in which the boat may be steered.

I drew a few diagrams, but nothing helped.
Original post by Mathematician99
Hi,

I am extremely stuck on Question 12 Exercise 1A in the Edexcel M4 textbook. The question is: A river flows at speed u. A boat is rowed with speed v relative to the river. The width of the river is w and the boat is to reach the opposite bank at a distance d downstream. Show that, if (uv/sqrt((w^2)+(d^2)))<v<u there are two directions in which the boat may be steered.

I drew a few diagrams, but nothing helped.
Firstly, I'm doubting the condition you've posted - unless I'm misunderstanding, any answer will only depend on the relative values of w, d, not the absolute values (because all that matters is the direction the boat goes in, not how *far* it goes).

Anyhow, what I would do is draw a diagram where the x, y axes represent velocity, not position.

So work out the direction that the boat needs to go, and draw the line of all points in that direction. (so. if the boat needs to go in the same direction as (1,2), you'd end up with the line of all +ve multiples of (1,2), which is just the line from the origin through (1,2) and onwards).

The direction that the boat *can* go will be a circle, offset by the origin by the river velocity.

I think you should be able to finish from there. (But as I said, I don't see how the given answer can be correct unless I'm missing something or you have misstated the problem).
Original post by DFranklin
Firstly, I'm doubting the condition you've posted


Wise decision.

Condition should be uww2+d2<v<u\displaystyle\frac{uw}{\sqrt{w^2+d^2}}< v<u for there to be two directions.
(edited 7 years ago)
Original post by ghostwalker
Wise decision.

Condition should be uww2+d2v<u\displaystyle\frac{uw}{\sqrt{w^2+d^2}}\leq v<u for there to be two directions.
PRSOM

[I have to confess this is one topic I never really "got" at A-level - I would generally be able to brute force a solution or move it to vector algebra and solve that way, but the neat geometric-type solutions generally elude me...]
Original post by DFranklin
...


On reflection, they should have been strict inequalities - amended post.
Original post by ghostwalker


Hello, I'm doing this question too. I'm also very stuck. I don't understand what the vector diagram that they have drawn is showing.Untitled2.jpg
Original post by student1856
Hello, I'm doing this question too. I'm also very stuck. I don't understand what the vector diagram that they have drawn is showing.


The second diagram is showing the two possible values for the velocity vector v, simultaneously. With either of them the velocity of the boat relative to the bank is u+v and it must be in the direction of the dotted arrow - going towards the point a distance d down on the far bank.
Original post by ghostwalker

Hi, I'm doing a similar question. A fielder can run in two direction to intercept a ball. Either θ or 180-θ if he runs in an obtuse angle. Wouldn't the red angle that I have labeled theta be just 180-θ. Which would be 139.68 instead of 124.68 that they wrote. I'm not entirely sure if they made a mistake or I'm wrong. Thanks in advance.

Untitled4.jpg
Original post by student1856
Hi, I'm doing a similar question. A fielder can run in two direction to intercept a ball. Either θ or 180-θ if he runs in an obtuse angle. Wouldn't the red angle that I have labeled theta be just 180-θ. Which would be 139.68 instead of 124.68 that they wrote. I'm not entirely sure if they made a mistake or I'm wrong. Thanks in advance.



Yes the red angle is 180-theta, and is the other value for theta.

BUT, the 124.68 is not refering to that angle, it is refering to the third angle in the triangle, which they are working out in order to use the sine rule. See attached:

Untitled.jpg

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