A-Level Edexcel Maths Mechanics Variable Acceleration - 1D: [In this question i and j are ho

[In this question i and j are horizontal unit vectors due east and due north respectively and position vectors are given relative to a fixed origin O.] Two cars P and Q are moving on straight horizontal roads with constant velocities - Edexcel - A-Level Maths Mechanics - Question 1 - 2016 - Paper 1

Question 1

[In this question i and j are horizontal unit vectors due east and due north respectively and position vectors are given relative to a fixed origin O.] Two cars P a... show full transcript

Worked Solution & Example Answer:[In this question i and j are horizontal unit vectors due east and due north respectively and position vectors are given relative to a fixed origin O.] Two cars P and Q are moving on straight horizontal roads with constant velocities - Edexcel - A-Level Maths Mechanics - Question 1 - 2016 - Paper 1

Step 1

Find the direction of motion of Q, giving your answer as a bearing to the nearest degree.

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Answer

To find the direction of motion of Q, we can start with its velocity vector:

$ext{Velocity of Q} = 20i - 5j$

The direction can be calculated using the tangent of the angle formed by the vector with the positive x-axis (east direction):

$an heta = \frac{\text{opposite}}{\text{adjacent}} = \frac{-5}{20}$

Thus, we find:

$\theta = \tan^{-1}\left(-\frac{5}{20}\right) = \tan^{-1}\left(-\frac{1}{4}\right)$

Calculating this gives:

$\theta \approx -14.04°$

Since this angle is measured clockwise from the east direction, the bearing is:

$ext{Bearing} = 360° + (-14.04°) = 345.96° ext{ or approximately } 346°.$

Step 2

Find an expression for p in terms of t.

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Answer

The position vector p can be expressed as:

$p = (400 + 15t)i + (20t)j.$

This shows that p depends linearly on time t.

Step 3

q in terms of t.

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Answer

For the position vector q, we know the initial position is 800j and the velocity of Q is $(20i - 5j)$ m s $^{-1}$ :

$q = (800 - 5t)j + (20t)i.$

Thus, q also varies with time t.

Step 4

Find the position vector of Q when Q is due west of P.

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Answer

For Q to be due west of P, their j-components must be equal:

$20t = 800 - 5t$

Rearranging gives:

\Rightarrow t = \frac{800}{25} = 32 \text{ seconds}.$$ Now substituting t back into the position vector of Q: $$q = (20(32)i + (800 - 5(32))j) = 640i + 640j.$$ Thus, the position vector of Q when it is due west of P is: $$q = 640i + 640j.$$

[In this question i and j are horizontal unit vectors due east and due north respectively and position vectors are given relative to a fixed origin O.] Two cars P and Q are moving on straight horizontal roads with constant velocities - Edexcel - A-Level Maths Mechanics - Question 1 - 2016 - Paper 1

Find the direction of motion of Q, giving your answer as a bearing to the nearest degree.

Find an expression for p in terms of t.

q in terms of t.

Find the position vector of Q when Q is due west of P.

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