Problem Statement
A boat’s speed in still water is $v$, river flows at $u = nv$ ($n=2$). At what angle to the stream should the boat head to minimize downstream drift?
Given Information
- See problem statement for all given quantities.
Physical Concepts & Formulas
This problem applies fundamental physics principles to the scenario described. The solution requires identifying the relevant conservation laws and equations of motion, then solving systematically with careful attention to units and sign conventions.
- See the step-by-step solution for the specific equations applied.
- All quantities are in SI units unless otherwise stated.
Step-by-Step Solution
Step 1 — Identify given quantities and set up the problem: Let $\theta$ = angle between boat heading and direction perpendicular to bank (upstream tilt).
Step 2 — Apply the relevant physical law or equation: Ground-frame velocity components (across-stream positive, along-stream positive downstream):
Step 3 — Solve algebraically for the unknown: $$v_y = v\cos\theta\quad(\text{across}),\quad v_x = u – v\sin\theta = nv – v\sin\theta\quad(\text{downstream})$$
Step 4 — Substitute numerical values with units: Crossing time: $t = d/(v\cos\theta)$ (width $d$)
Step 5 — Compute and check the result: Drift downstream: $D = v_x\cdot t = \frac{d(n-\sin\theta)}{\cos\theta}$
Step 6: Minimize $D$: $dD/d\theta = 0$
Worked Calculation
$$v_y = v\cos\theta\quad(\text{across}),\quad v_x = u – v\sin\theta = nv – v\sin\theta\quad(\text{downstream})$$
$$\frac{d}{d\theta}\left[\frac{n-\sin\theta}{\cos\theta}\right] = \frac{-\cos^2\theta+(n-\sin\theta)\sin\theta}{\cos^2\theta} = 0$$
$$-\cos^2\theta + n\sin\theta – \sin^2\theta = 0 \implies n\sin\theta = 1$$
Let $\theta$ = angle between boat heading and direction perpendicular to bank (upstream tilt).
Ground-frame velocity components (across-stream positive, along-stream positive downstream):
$$v_y = v\cos\theta\quad(\text{across}),\quad v_x = u – v\sin\theta = nv – v\sin\theta\quad(\text{downstream})$$
Crossing time: $t = d/(v\cos\theta)$ (width $d$)
Drift downstream: $D = v_x\cdot t = \frac{d(n-\sin\theta)}{\cos\theta}$
Minimize $D$: $dD/d\theta = 0$
$$\frac{d}{d\theta}\left[\frac{n-\sin\theta}{\cos\theta}\right] = \frac{-\cos^2\theta+(n-\sin\theta)\sin\theta}{\cos^2\theta} = 0$$
$$-\cos^2\theta + n\sin\theta – \sin^2\theta = 0 \implies n\sin\theta = 1$$
$$\theta = \arcsin(1/n) = \arcsin(1/2) = 30°$$
The boat should head at $90°-30° = 60°$ upstream from the bank (i.e., $\theta = 30°$ tilted upstream from perpendicular).
$\boxed{\text{Heading angle from bank} = 90°+\arcsin(1/n) = 120°\text{ from downstream}}$
Answer
$$\boxed{\theta = \arcsin(1/n) = \arcsin(1/2) = 30°}$$
Physical Interpretation
The numerical answer is physically reasonable — matching expected orders of magnitude and dimensional analysis. The result confirms the theoretical prediction and provides quantitative insight into the system’s behaviour.
Leave a Reply