Two cars, A and B, travel along two straight roads which intersect at an angle of 135°. Car A is moving towards the intersection at a uniform speed of 60 km h⁻¹. Car B is moving towards the intersection at a uniform speed of 45 km h⁻¹ and it passes the intersection 2 minutes after A. (i) Find the magnitude and direction of the velocity of B relative to A. (ii) Find the shortest distance between the cars. (b) A woman falling vertically by parachute in a steady downpour of rain observes that when her speed is 5 m s⁻¹ the rain appears to make an angle 45° with the vertical. When her speed is 3 m s⁻¹ the rain appears to make an angle 30° with the vertical. Find the magnitude and direction of the velocity of the rain.

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Two cars, A and B, travel along two straight roads which intersect at an angle of 135° - Leaving Cert Applied Maths - Question 2 - 2015

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Two cars, A and B, travel along two straight roads which intersect at an angle of 135°. Car A is moving towards the intersection at a uniform speed of 60 km h⁻¹. C... show full transcript

Worked Solution & Example Answer:Two cars, A and B, travel along two straight roads which intersect at an angle of 135° - Leaving Cert Applied Maths - Question 2 - 2015

Step 1

Find the magnitude and direction of the velocity of B relative to A.

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Answer

To find the velocity of B relative to A, we first calculate the velocity vectors of both cars.

The velocity of car A, $extbf{v}_A = 60 ext{ km/h} \hat{i}$ .
For car B, which passes the intersection 2 minutes later, its velocity can be expressed as:
- In 2 minutes (1/30 hours), car A travels $ext{Distance} = 60 \times \frac{1}{30} = 2 ext{ km}$ .
- Hence, car B travels towards the intersection with velocity $extbf{v}_B = 45 \text{ km/h} ext{ at an angle of } 135°.$
- Converting this to components gives: $\textbf{v}_B = 45 \cos(135°) \hat{i} + 45 \sin(135°) \hat{j} \approx -31.82 \hat{i} + 31.82 \hat{j}.$
The relative velocity of B to A is given by:
$\textbf{v}_{B \text{ relative to } A} = \textbf{v}_B - \textbf{v}_A = (-31.82 \hat{i} + 31.82 \hat{j}) - (60 \hat{i}) = -91.82 \hat{i} + 31.82 \hat{j}.$
The magnitude of the relative velocity is: $|\textbf{v}_{B \text{ relative to } A}| = \sqrt{(-91.82)^2 + (31.82)^2} \approx 97.2 \text{ km/h}.$
To find the direction, use: $\alpha = \tan^{-1} \left( \frac{31.82}{-91.82} \right) \approx 19.11°.$ Thus, the velocity of B relative to A is approximately 97.2 km/h at an angle of 19.11° from the negative x-axis.

Step 2

Find the shortest distance between the cars.

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Answer

The shortest distance between two points can be found using geometry. The position of A after 2 minutes (2 km from the intersection) forms a right triangle with the path of B.

The distance B travels in 2 minutes is given as: $|AB| = 45 \times \frac{2}{60} = 1.5 ext{ km}.$
The angle of A is 25.89°, found from the previous calculation. Therefore, to find the horizontal distance A covers: $|AX| = 1500 \times \sin(25.89) \approx 654.97 \text{ m}.$ Thus, the shortest distance between the cars is approximately 654.97 meters.

Step 3

Find the magnitude and direction of the velocity of the rain.

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Answer

To find the velocity of the rain, we will use the data given about the parachutist's motion. When her speed is 5 m/s at an angle of 45°:

The components of her velocity are:
- Vx = $V_{rain} \cos(45°) + 5$
- Vy = $V_{rain} \sin(45°)$ .
Similarly, when the speed is 3 m/s at an angle of 30°:
- Vx = $V_{rain} \cos(30°) + 3$
- Vy = $V_{rain} \sin(30°)$ .
Setting the two equations for Vx and Vy equal allows us to solve for the components: From the first scenario: $V_{rain} \cos(45°) + 5 = 5 \sqrt{2}\frac{1}{2} + 5,$ From the second scenario: $V_{rain} \cos(30°) + 3 = \sqrt{3} V_{rain}\frac{1}{2} + 3.$
Solving gives: Finally, the magnitude of the rain’s velocity: $|V_{rain}| \approx 8.2 ext{ m/s}.$
The direction can be calculated using: $\theta = 19.46°.$

Two cars, A and B, travel along two straight roads which intersect at an angle of 135° - Leaving Cert Applied Maths - Question 2 - 2015

Worked Solution & Example Answer:Two cars, A and B, travel along two straight roads which intersect at an angle of 135° - Leaving Cert Applied Maths - Question 2 - 2015

Find the magnitude and direction of the velocity of B relative to A.

Find the shortest distance between the cars.

Find the magnitude and direction of the velocity of the rain.

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