A-Level Edexcel Maths Mechanics Constant 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 need to determine the angle of the velocity vector of Q, which is given as $(20i - 5j)$ m s $^{-1}$ . The direction can be calculated using the tangent function:

$\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{-5}{20}$

Calculating this gives:

$\theta = \tan^{-1}\left(-\frac{5}{20}\right) = -14.036^{\circ}$

To convert this to a bearing, we must adjust for the quadrant. Since the vector is in the fourth quadrant, the bearing can be calculated as:

$q = 360^{\circ} + (-14.036^{\circ}) = 345.964^{\circ} \approx 346^{\circ}$

Thus, the bearing is 346°.

Step 2

Find an expression for p in terms of t.

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Answer

The position vector of P at time $t$ can be defined based on its initial position and its velocity:

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

Step 3

Find an expression for q in terms of t.

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Answer

Similarly, the position vector of Q can be defined as follows:

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

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 i-components must be equal in magnitude but opposite in sign. From the expressions derived:

$\text{Set } (400 + 15t) = -(20t)$

Solving this gives:

$400 + 15t + 20t = 0 \implies 400 + 35t = 0 \implies t = -\frac{400}{35} \implies t = -\frac{80}{7}$

This indicates that we have a prior time when this condition is satisfied. Thus, substituting this back into the expression for q gives the position vector:

[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.

Find an expression for q in terms of t.

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

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