Question 9: Motion in 2D & Vectors

Question

A river of width 200 m is flowing from west to east with a speed of 18 km/h. A boat, moving with speed of 36 km/h in still water, is made to travel one-round trip (bank to bank of the river). Minimum time taken by the boat for this journey and also the displacement along the river bank are ______ and ______ respectively.

Accepted Answer

The speed of river is $v_r=18 	imes \frac{5}{18}=5 \mathrm{~m} / \mathrm{s}$
The speed of boat in still water is $\mathrm{v}_{\mathrm{b}}=36 	imes \frac{5}{18}=10 \mathrm{~m} / \mathrm{s}$
The width of river is $\mathrm{w}=200 \mathrm{~m}$
The time taken to cross a river is determined solely by the component of the boat's velocity perpendicular to the river banks.
$$ t=\frac{w}{v_b \sin 	heta} $$
Where theta is the angle with the bank. Time is minimum when the boat heads straight across ( $	heta=90^{\circ}$ ), so $\sin 	heta=1$.
$$ t_{\min 1 ightarrow 2}=\frac{w}{v_b}=\frac{200}{10}=20 \mathrm{~s} $$
Also, while returning to initial bank in minimum time,
$$ t_{\min 2 ightarrow 1}=\frac{w}{v_b}=\frac{200}{10}=20 \mathrm{~s} $$
Total time of journey is,
$$ \mathrm{T}_{	ext {total }}=\mathrm{t}_{\min 1 ightarrow 2}+\mathrm{t}_{\min 2 ightarrow 1}=2 	imes 20=40 \mathrm{~s} $$
When the boat heads straight across to minimize time, the river flow carries it downstream. This horizontal distance is,
Drift (one way) $=\mathrm{v}_{\mathrm{r}} 	imes \mathrm{t}_{	ext {min } 1 ightarrow 2}=5 	imes 20 \mathrm{~s}=100 \mathrm{~m}$ east
The boat again heads straight across to minimize time. The river still flows from west to east, so the boat drifts another 100 m east.
Total Displacement along the bank:
$$ \Delta \mathrm{x}=100 \mathrm{~m} 	ext { (east) }+100 \mathrm{~m} 	ext { (east) }=200 \mathrm{~m} $$
Therefore, the minimum time is 40 s and the displacement along the bank is 200 m . Hence, the correct option is (C).

Answer

20 s and 100 m

Answer

40 s and 100 m

Answer

40 s and 200 m

Answer

40 s and 0 m

Motion in 2D & Vectors

Choose the correct answer: