Problem Statement
Solve the kinematics problem: A man walking at $v_1 = 3\,\text{m/s}$ in rain that falls vertically at $v_2 = 6\,\text{m/s}$. At what angle to the vertical does the rain appear to fall relative to the man? How fast does the apparent rain move? Rain velocity in ground frame: $\vec v_{rain} = -v_2\hat j = -6\hat j\,\text{m/s}$ Man’
Given Information
- $v_1 = 3\,\text{m/s}$
- $v_2 = 6\,\text{m/s}$
Physical Concepts & Formulas
Relative velocity $\mathbf{v}_{AB} = \mathbf{v}_A – \mathbf{v}_B$ is the velocity of A as seen from B’s reference frame. For river-crossing problems, the boat’s velocity relative to ground = boat velocity relative to water + water velocity. The angle for minimum crossing time differs from the angle for minimum drift.
- $\mathbf{v}_{A/B} = \mathbf{v}_A – \mathbf{v}_B$
- River crossing: $\mathbf{v}_{\text{ground}} = \mathbf{v}_{\text{boat/water}} + \mathbf{v}_{\text{river}}$
- Minimum time: boat aims perpendicular to river bank
- Minimum drift: $\sin\phi = v_{\text{boat}}/v_{\text{river}}$ (if river faster)
Step-by-Step Solution
Step 1 — Verify the result: Check units, limiting cases, and order of magnitude to confirm the answer is physically reasonable.
Step 2 — Verify the result: Check units, limiting cases, and order of magnitude to confirm the answer is physically reasonable.
Step 3 — Verify the result: Check units, limiting cases, and order of magnitude to confirm the answer is physically reasonable.
Worked Calculation
$$R = \frac{u^2\sin 2\theta}{g} = \frac{400\times\sin 60°}{9.8} = \frac{400\times0.866}{9.8} = \frac{346.4}{9.8} \approx 35.3\,\text{m}$$
$$H = \frac{u^2\sin^2\theta}{2g} = \frac{400\times0.25}{19.6} = \frac{100}{19.6} \approx 5.1\,\text{m}$$
$$\boxed{R = \dfrac{u^2\sin 2\theta}{g},\quad H = \dfrac{u^2\sin^2\theta}{2g}}$$
Answer
$$\boxed{R = \dfrac{u^2\sin 2\theta}{g},\quad H = \dfrac{u^2\sin^2\theta}{2g}}$$
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