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Two straight roads intersect at an angle of 30° - Leaving Cert Applied Maths - Question 2 - 2020
Question 2
Two straight roads intersect at an angle of 30°. Car A is moving along one road towards the intersection with a uniform speed of 6 m s⁻¹. Car B is moving along the ... show full transcript
Worked Solution & Example Answer:Two straight roads intersect at an angle of 30° - Leaving Cert Applied Maths - Question 2 - 2020
Step 1
Find the velocity of B relative to A.
Answer
To find the velocity of car B relative to car A, we can use the formula:
Car A's velocity is directed along the road at 6 m s⁻¹:
Car B's velocity can be resolved into components:
- The angle given is 30 degrees, which means: Plugging in the values: extbf{V}_B = 8 imes rac{ ext{√3}}{2} extbf{i} + 8 imes rac{1}{2} extbf{j} = 4 ext{√3} extbf{i} + 4 extbf{j}
Now, substituting these into the relative velocity equation:
Calculating the magnitude gives:
= ext{√}igg( (4 ext{√3} - 6)² + 16igg)$$ After calculation, this results in approximately 4.1 m s⁻¹.Step 2
Find the distance of each car from the intersection when they are nearest to each other.
Answer
To find the distance when the cars are nearest, we first find the time taken for car A to reach the intersection:
Assuming is the distance of car B from the intersection at this point, we know that:
Since car A arrives 5 seconds before car B, we can say: Thus, substituting:
Now, for the next car’s distance: Using the angle and the time (6 m for A, needing to compute time for B according to B's speed and time): Using similar calculations yields: .
Step 3
Find x in terms of d.
Answer
Applying the swimmer's journey, we set up equations:
Vertical distance (across the river):
rac{d}{1.5} + t_c = d
where is time spent going downstream.
Horizontal distance (downstream due to current):
rac{16}{2} = d
And also:
t_d = rac{2x}{d}
Both components give proper relationships to reach: Finally we arrange to isolate x: x = rac{2d}{2} = d leading to: for your calculations.