Two ships P and Q are moving along straight lines with constant velocities. Initially P is at a point O and the position vector of Q relative to O is (6i + 12j) km, where i and j are unit vectors directed due east and due north respectively. The ship P is moving with velocity 10j km h⁻¹ and Q is moving with velocity (-8i + 6j) km h⁻¹. At time t hours the position vectors of P and Q relative to O are p km and q km respectively. (a) Find p and q in terms of t. (b) Calculate the distance of Q from P when t = 3. (c) Calculate the value of t when Q is due north of P.

A-Level Edexcel Maths Mechanics Working with Vectors: Two ships P and Q are moving alo

Two ships P and Q are moving along straight lines with constant velocities - Edexcel - A-Level Maths Mechanics - Question 4 - 2003 - Paper 1

Question 4

Two ships P and Q are moving along straight lines with constant velocities. Initially P is at a point O and the position vector of Q relative to O is (6i + 12j) km, ... show full transcript

Worked Solution & Example Answer:Two ships P and Q are moving along straight lines with constant velocities - Edexcel - A-Level Maths Mechanics - Question 4 - 2003 - Paper 1

Step 1

Find p and q in terms of t.

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Answer

The position vector of ship P at time t hours is given by:

extbf{p} = 10t j ext{ km}

The position vector of ship Q at time t hours is calculated by considering its initial position and its motion:

extbf{q} = (6i + 12j) + t(-8i + 6j) = (6 - 8t)i + (12 + 6t)j ext{ km}

Step 2

Calculate the distance of Q from P when t = 3.

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Answer

Substituting t = 3:

extbf{p} = 10(3)j = 30j ext{ km}

For Q:

extbf{q} = (6 - 8(3))i + (12 + 6(3))j = (6 - 24)i + (12 + 18)j = -18i + 30j ext{ km}

The distance PQ can be calculated using the formula:

ext{Distance} = || extbf{q} - extbf{p}||

Calculating PQ:

extbf{PQ} = (-18i + 30j) - (0i + 30j) = -18i ext{ km}

The distance is:

| extbf{PQ}| = | -18 | = 18 ext{ km}

Step 3

Calculate the value of t when Q is due north of P.

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Answer

For Q to be due north of P, the eastward component of Q must equal zero. This occurs when:

6 - 8t = 0

Solving for t gives:

8t = 6 \ t = rac{6}{8} = rac{3}{4} ext{ hours}

Two ships P and Q are moving along straight lines with constant velocities - Edexcel - A-Level Maths Mechanics - Question 4 - 2003 - Paper 1

Worked Solution & Example Answer:Two ships P and Q are moving along straight lines with constant velocities - Edexcel - A-Level Maths Mechanics - Question 4 - 2003 - Paper 1

Find p and q in terms of t.

Calculate the distance of Q from P when t = 3.

Calculate the value of t when Q is due north of P.

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