A river is 72 metres wide and has parallel banks. A boat B departs from point p on the southern bank and lands at point r on the northern bank. The actual velocity of B is $-4oldsymbol{i} + 3oldsymbol{j}$ m/s. Cyclist C travels due north at a constant speed of 4 m/s across a straight level bridge which spans the river. Find (i) the velocity of C in terms of $\boldsymbol{i}$ and $\boldsymbol{j}$ (ii) the velocity of B relative to C in terms of $\boldsymbol{i}$ and $\boldsymbol{j}$ (iii) the magnitude and direction of the velocity of B relative to C (iv) the time it takes C to cross the river (v) how much longer it will take B to cross the river.

Leaving Cert Applied Maths Relative Velocity: A river is 72 metres wide and ha

A river is 72 metres wide and has parallel banks - Leaving Cert Applied Maths - Question 2 - 2007

Question 2

A river is 72 metres wide and has parallel banks. A boat B departs from point p on the southern bank and lands at point r on the northern bank. The actual velocity ... show full transcript

Worked Solution & Example Answer:A river is 72 metres wide and has parallel banks - Leaving Cert Applied Maths - Question 2 - 2007

Step 1

the velocity of C in terms of $\boldsymbol{i}$ and $\boldsymbol{j}$

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Answer

$\vec{v}_C = 0\boldsymbol{i} + 4\boldsymbol{j}$ m/s

Step 2

the velocity of B relative to C in terms of $\boldsymbol{i}$ and $\boldsymbol{j}$

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Answer

$\vec{v}_{BC} = \vec{v}_B - \vec{v}_C = (-4\boldsymbol{i} + 3\boldsymbol{j}) - (0\boldsymbol{i} + 4\boldsymbol{j}) = -4\boldsymbol{i} - 1\boldsymbol{j}$

Step 3

the magnitude and direction of the velocity of B relative to C

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Answer

Magnitude: $\sqrt{(-4)^2 + (-1)^2} = \sqrt{16 + 1} = \sqrt{17} \approx 4.12$ m/s. Direction: $\tan^{-1}\left(\frac{-1}{-4}\right) \approx 14.04^\circ$ with bank.

Step 4

the time it takes C to cross the river

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Answer

Time: $\frac{72}{4} = 18$ seconds.

Step 5

how much longer it will take B to cross the river

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Answer

Time: $\frac{72}{3} = 24$ seconds. Thus, required time difference: $24 - 18 = 6$ seconds.

A river is 72 metres wide and has parallel banks - Leaving Cert Applied Maths - Question 2 - 2007

Worked Solution & Example Answer:A river is 72 metres wide and has parallel banks - Leaving Cert Applied Maths - Question 2 - 2007

the velocity of C in terms of $\boldsymbol{i}$ and $\boldsymbol{j}$

the velocity of B relative to C in terms of $\boldsymbol{i}$ and $\boldsymbol{j}$

the magnitude and direction of the velocity of B relative to C

the time it takes C to cross the river

how much longer it will take B to cross the river

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