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Two straight roads cross at right angles - Leaving Cert Applied Maths - Question 2 - 2008
Question 2
Two straight roads cross at right angles. A woman C, is walking towards the intersection with a uniform speed of 1.5 m/s. Another woman D is moving towards the int... show full transcript
Worked Solution & Example Answer:Two straight roads cross at right angles - Leaving Cert Applied Maths - Question 2 - 2008
Step 1
Find (i) the velocity of C relative D
Answer
To find the velocity of C relative to D ( ( \mathbf{v}_{C/D} )), we can use the following formula:
[ \mathbf{v}_{C/D} = \mathbf{v}_C - \mathbf{v}_D ]
Where:
- ( \mathbf{v}_C = 1.5 \mathbf{j} ) (C is moving towards the North)
- ( \mathbf{v}_D = 0\mathbf{i} + 2\mathbf{j} ) (D is moving towards the South)
Calculating the relative velocity:
[ \mathbf{v}_{C/D} = (1.5 \mathbf{j}) - (0 + 2\mathbf{j}) = -0.5\mathbf{j} ]
Magnitude of ( \mathbf{v}_{C/D} ):
[ \text{Magnitude} = |\mathbf{v}_{C/D}| = 2.5 , \text{m/s} ]
Direction is calculated by:
[ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{1.5}{2} \text{ (East 53.13° South)} ]
Step 2
Find (ii) the distance of C from the intersection when they are nearest together
Answer
To find the distance of C from the intersection when they are nearest:
- First, calculate the time ( t ):
[ t = \frac{|\mathbf{C \to D}|}{|\mathbf{v}_{C/D}| \cos(53.13°)} = \frac{100}{2.5} = 24 \text{ s} ]
- Then calculate the distance C travels during this time:
[ \text{Distance} = 1.5 \times 24 = 36 \text{ m} ]
- The distance of C from the intersection is:
[ 100 - 36 = 64 , \text{m} ]
Step 3
On a particular day the velocity of the wind, in terms of i and j, is x i - 3 j, where x is N. Find the actual direction of the wind.
Answer
Given the wind's velocity ( \mathbf{v} = x \mathbf{i} - 3 \mathbf{j} ):
- Set ( \mathbf{v_{max}} = a \mathbf{i} ):
[ \mathbf{v_{man}} = \mathbf{v} - \mathbf{v_{max}} ]
- For direction North ( \alpha ):
[ \tan(\alpha) = \frac{x - a}{3} \Rightarrow a = x - 6 ]
- Hence, calculate for West direction ( \beta ):
[ \tan(\beta) = \frac{3}{\frac{2}{3} (3 + 4)} \Rightarrow \text{final calculations yield actual wind direction} = 71.6° \text{ West} ]