A.1 · 5 · A.1 Kinematics · vectors and scalars
Vectors: components and the river
Steer a boat across a flowing river, then learn to resolve any vector into two parts.
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Point a boat straight across a flowing river. Does it land on the opposite bank, or somewhere downstream?
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A vector of magnitude 8 points at 60° above the horizontal. Its vertical component is…
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A boat aims straight across a river (θ = 0°). Compared with aiming slightly upstream, its crossing time is…
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A river flows at 2 m s⁻¹. A boat that can only manage 1.5 m s⁻¹…
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To rebuild a vector from its components Vx and Vy, its magnitude is…
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A river is 30 m wide and flows at 1.5 m s⁻¹. A boat can travel at 2.5 m s⁻¹ in still water. If the boat aims straight across, find the time to cross and the drift downstream.
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Remember earlier lessons
From lesson A.1 · 1
Motion is described…
From lesson A.1 · 4
Uniform motion means…
Summary card
Key points
- • A vector resolves into components: Vx = V cos θ, Vy = V sin θ.
- • When two velocities act at once, the real motion is their vector sum.
- • Crossing a river: only the across component gets you over; the along component (current + boat) causes drift.
Formulas
- Vx = V cos θ, Vy = V sin θ
- V = √(Vx² + Vy²)
- across = V cos θ, along = current + V sin θ
Key terms
- Resolve :
- split a vector into two components at right angles
- Resultant :
- the single vector that has the same effect as two or more vectors added together
Common errors
- • Mixing up sin and cos for the two components.
- • Forgetting the current still acts even when the boat aims straight across.
- • Assuming you can always cancel the drift: only true if the boat is faster than the current.
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