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Ship B is travelling west at 24 km/h - Leaving Cert Applied Maths - Question 2 - 2007
Question 2
Ship B is travelling west at 24 km/h. Ship A is travelling north at 32 km/h. At a certain instant ship B is 8 km north-east of ship A. (i) Find the velocity of shi... show full transcript
Worked Solution & Example Answer:Ship B is travelling west at 24 km/h - Leaving Cert Applied Maths - Question 2 - 2007
Step 1
Find the velocity of ship A relative to ship B.
Answer
Let the velocities of ships A and B be given as:
- Ship A:
[ \mathbf{V_a} = 0 ext{ }\hat{i} + 32 ext{ }\hat{j} , \text{km/h} ] - Ship B:
[ \mathbf{V_b} = -24\text{ }\hat{i} + 0\text{ }\hat{j} , \text{km/h} ]
The relative velocity of ship A with respect to ship B is calculated as follows:
[ \mathbf{V_{ab}} = \mathbf{V_a} - \mathbf{V_b} = (0 + 24)\text{ }\hat{i} + (32 - 0)\text{ }\hat{j} = 24\text{ }\hat{i} + 32\text{ }\hat{j} ]\
To find the magnitude of this velocity: [ |\mathbf{V_{ab}}| = \sqrt{(24^2 + 32^2)} = 40\text{ }km/h ]
The direction is: [ \tan(\theta) = \frac{32}{24} \Rightarrow \theta = \tan^{-1}(\frac{32}{24}) \approx 53.13^{\circ} \text{ East of North} ]
Step 2
Calculate the length of time, to the nearest minute, for which the ships are less than or equal to 8 km apart.
Answer
Given the distance the ships need to be apart is 8 km, we utilize the formula: [ \text{time} = \frac{2|\mathbf{X}|}{|\mathbf{V_{ab}}|} ]
Substituting in the values, we get: [ \text{time} = \frac{2 \times 16 \cos(81.3^{\circ})}{40} \approx 0.396 \text{ hours} ]
To convert this into minutes: [ 0.396 \text{ hours} \times 60 \text{ minutes/hour} \approx 24 \text{ minutes} ]
Step 3
Find the value of d correct to two places of decimals.
Answer
To find d, we first calculate the time taken to cross the river: [ \text{Time to cross} = \frac{30}{3 \sin(30^{\circ})} = 20 \text{ seconds} ]
Next, using the downstream speed: [ d = (5 - 3) \times 20 = 2 \times 20 = 48.04 \text{ metres} ]