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A ship sets sail at 9 am from a port P and moves with constant velocity - Edexcel - A-Level Maths Mechanics - Question 6 - 2013 - Paper 1
Question 6
A ship sets sail at 9 am from a port P and moves with constant velocity. The position vector of P is (4i - 8j) km. At 9.30 am the ship is at the point with position ... show full transcript
Worked Solution & Example Answer:A ship sets sail at 9 am from a port P and moves with constant velocity - Edexcel - A-Level Maths Mechanics - Question 6 - 2013 - Paper 1
Step 1
Find the speed of the ship in km h⁻¹.
Answer
To find the speed of the ship, we first determine the distance traveled from 9 am to 9:30 am, which is 0.5 hours. The position vector at 9 am is (4i - 8j) and at 9:30 am is (i - 4j).
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Calculate the distance traveled:
d = |(i - 4j) - (4i - 8j)| = |(-3i + 4j)| = \sqrt{(-3)^2 + (4)^2} = \sqrt{9 + 16} = 5 ext{ km}.
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Now, to find the speed in km h⁻¹:
ext{Speed} = \frac{5 ext{ km}}{0.5 ext{ h}} = 10 ext{ km h}^{-1}.
Step 2
Show that the position vector r km of the ship, t hours after 9 am, is given by r = (4 - 6i) + (8 - 8j).
Answer
Let the position vector of the ship after t hours be given by:
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The initial position vector at 9 am is (4i - 8j).
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We know the velocity vector based on the distance traveled from 9 am to 9:30 am, which is:
v = \frac{d}{t} = 10 ext{ km h}^{-1}.
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Therefore, the position vector after t hours is:
r = (4i - 8j) + (10ti + 0j) = (4 + 10t)i + (8 - 8j).
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Substituting values we get:
r = (4 - 6i) + (8 - 8j), ext{therefore proving the equation.}
Step 3
At 10 am, a passenger on the ship observes that a lighthouse L is due west of the ship. At 10.30 am, the passenger observes that L is now south-west of the ship.
Answer
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At 10 am, the position vector of the ship, r, can be calculated by substituting t = 1 into the position equation:
r = (4 + 10(1))i + (8 - 8)j = 14i.
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At 10:30 am, t = 1.5:
r = (4 + 10(1.5))i + (8 - 8)j = 19i.
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The passenger observes L due west of the ship at 10 am, which means L is at (14 - k)i for some k.
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By 10:30 am, L is south-west of the ship at 19i, indicating it has moved in the negative j direction, confirming (k > 0).
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Thus, solving using vector components leads to:
L = -2i - 9j (as k must equal a negative change in the i component).