[In this question, the unit vectors i and j are due east and due north respectively. Position vectors are relative to a fixed origin O.] A boat P is moving with constant velocity (−4i+8j) km h⁻¹. (a) Calculate the speed of P. When t = 0, the boat P has position vector (2i−8j) km. At time t hours, the position vector of P is p km. (b) Write down p in terms of t. A second boat Q is also moving with constant velocity. At time t hours, the position vector of Q is q km, where q = 18i + 12j − (6i + 8j) Find (c) the value of t when P is due west of Q, (d) the distance between P and Q when P is due west of Q.

A-Level Edexcel Maths Mechanics Constant Acceleration - 1D: [In this question, the unit vect

[In this question, the unit vectors i and j are due east and due north respectively - Edexcel - A-Level Maths Mechanics - Question 7 - 2012 - Paper 1

Question 7

[In this question, the unit vectors i and j are due east and due north respectively. Position vectors are relative to a fixed origin O.] A boat P is moving with c... show full transcript

Worked Solution & Example Answer:[In this question, the unit vectors i and j are due east and due north respectively - Edexcel - A-Level Maths Mechanics - Question 7 - 2012 - Paper 1

Step 1

Calculate the speed of P.

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Answer

To find the speed of boat P, we use the formula for speed given by the magnitude of the velocity vector:
$\text{Speed} = \sqrt{(-4)^2 + (8)^2} \text{ km h}^{-1}$
This simplifies to:
$\text{Speed} = \sqrt{16 + 64} = \sqrt{80} \text{ km h}^{-1}$
Thus, the speed of P is approximately 8.94 km h⁻¹.

Step 2

Write down p in terms of t.

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Answer

The position vector p of boat P as it moves over time can be expressed as:
p = (2i - 8j) + t(-4i + 8j)
This gives:
p = [2 - 4t]i + [-8 + 8t]j.

Step 3

Find the value of t when P is due west of Q.

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Answer

Boat P is due west of boat Q when their respective j-components are equal.
From the position vector of Q:
q = (18 - 6)i + (12 - 8)j
This can be simplified to:
q = 12i + 4j.
Setting the j-components equal gives:
$-8 + 8t = 4$
Solving for t:
$8t = 12$
$t = \frac{12}{8} = \frac{3}{2} = 1.5 \text{ hours}$ .

Step 4

Find the distance between P and Q when P is due west of Q.

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Answer

Using the calculated value of t from part (c), we substitute t = 1.5 into the position vector for P:
p = [2 - 4(1.5)]i + [-8 + 8(1.5)]j
This simplifies to:
p = [-4]i + [4]j.
Now we can find the position of Q at t = 1.5:
q = 12i + 4j.
To find the distance between P and Q:
$\text{Distance} = \sqrt{((-4 - 12)^2 + (4 - 4)^2)} = \sqrt{(-16)^2} = 16 \text{ km}.$

[In this question, the unit vectors i and j are due east and due north respectively - Edexcel - A-Level Maths Mechanics - Question 7 - 2012 - Paper 1

Worked Solution & Example Answer:[In this question, the unit vectors i and j are due east and due north respectively - Edexcel - A-Level Maths Mechanics - Question 7 - 2012 - Paper 1

Calculate the speed of P.

Write down p in terms of t.

Find the value of t when P is due west of Q.

Find the distance between P and Q when P is due west of Q.

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