Problem Statement
Solve the kinematics problem: Particle 1 moves along $x$-axis with constant acceleration $a$ from rest. Particle 2 moves along $y$-axis with constant velocity $b$. Find the angle $\theta$ that the velocity of particle 2 relative to particle 1 makes with the $x$-axis at time $t$. Velocities: $\vec v_1 = at\hat x$, $\vec v_2 = b\h
Given Information
- See problem statement for all given quantities.
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
Full substitution shown in the steps above.
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.
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