Two cars, A and B, travel along two straight roads which intersect at an angle $ heta$. Car A is moving towards the intersection at a uniform speed of 9 m s$^{-1}$. Car B is moving towards the intersection at a uniform speed of 15 m s$^{-1}$. At a certain instant each car is 90 m from the intersection and approaching the intersection. (i) Find the distance between the cars when B is at the intersection. (ii) If the shortest distance between the cars is 36 m, find the value of $ heta$. An aircraft P, flying at 600 km h$^{-1}$, sets out to intercept a second aircraft Q, which is a distance away in a direction west 30° south, and flying due east at 600 km h$^{-1}$. Find the direction in which P should fly in order to intercept Q.

Leaving Cert Applied Maths Relative Velocity: Two cars, A and B, travel along

Two cars, A and B, travel along two straight roads which intersect at an angle $ heta$ - Leaving Cert Applied Maths - Question 2 - 2013

Question 2

Two cars, A and B, travel along two straight roads which intersect at an angle $ heta$. Car A is moving towards the intersection at a uniform speed of 9 m s$^{-1}$.... show full transcript

Worked Solution & Example Answer:Two cars, A and B, travel along two straight roads which intersect at an angle $ heta$ - Leaving Cert Applied Maths - Question 2 - 2013

Step 1

Find the distance between the cars when B is at the intersection.

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Answer

To find the distance between cars A and B when B reaches the intersection, we can use the Pythagorean theorem. Each car is 90 m away from the intersection, so the distance between them can be calculated as:

$|AB| = \sqrt{90^2 + 90^2} = \sqrt{8100} = 90 \sqrt{2} \approx 127.28 m$

Thus, the distance between the two cars is approximately 127.28 m.

Step 2

If the shortest distance between the cars is 36 m, find the value of θ.

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Answer

The relationship can be established using the sine rule, given that the speeds are different. The speed of car A is 9 m s $^{-1}$ and that of car B is 15 m s $^{-1}$ , thus:

$\frac{|AB|}{\sin(\theta)} = \frac{90}{15} \Rightarrow \sin(\theta) = \frac{9}{15} = 0.6$

This gives:

$\theta = \arcsin(0.6) \approx 36.87°$

Step 3

Find the direction in which P should fly in order to intercept Q.

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Answer

For aircraft P to intercept Q:

Let the velocity vector of P be represented as $extbf{V}_{p} = 600 \cos(a) \hat{i} - 600 \sin(a) \hat{j}$ where $a$ is the angle to be determined.
The velocity vector of Q is $extbf{V}_{q} = 600 \hat{i}$ . Therefore, the relative velocity vector is: $\textbf{V}_{r} = \textbf{V}_{p} - \textbf{V}_{q} = (600 \cos(a) - 600) \hat{i} - 600 \sin(a) \hat{j}$
To determine the intercept line, set up: $\tan(30°) = \frac{600 \sin(a)}{600(1 - \cos(a))}$
Simplifying leads to: $\sin(a) = \frac{1}{2}, \therefore a = 30°$
Thus, P should head $W 60° S$ or $S 30° W$ .

Two cars, A and B, travel along two straight roads which intersect at an angle $ heta$ - Leaving Cert Applied Maths - Question 2 - 2013

Worked Solution & Example Answer:Two cars, A and B, travel along two straight roads which intersect at an angle $ heta$ - Leaving Cert Applied Maths - Question 2 - 2013

Find the distance between the cars when B is at the intersection.

If the shortest distance between the cars is 36 m, find the value of θ.

Find the direction in which P should fly in order to intercept Q.

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