A.1 · IB Physics
Relative velocity: crossing a river
Steer a boat across a 60.0 m wide river with a current. Watch how the boat's velocity relative to the water adds to the current's velocity, and see how the heading changes the crossing time and the downstream drift.
boat relative to water current resultant (relative to bank)
v⃗(boat, bank) = v⃗(boat, water) + v⃗(water, bank)
vₓ = u − v sinθ ; v_y = v cosθ
t = d ÷ (v cosθ)
drift = (u − v sinθ) × t
straight across: sinθ = u ÷ v
vₓ = u − v sinθ ; v_y = v cosθ
t = d ÷ (v cosθ)
drift = (u − v sinθ) × t
straight across: sinθ = u ÷ v
Try this
- Keep θ = 0° and raise the current from 0 to 3 m s⁻¹. Does the crossing time change? What happens to the drift?
- Set v = 3.0 m s⁻¹, u = 2.0 m s⁻¹ and tick 'aim straight across'. Check that θ = 41.8° and compare the crossing time with heading straight across.
- Make u larger than v and tick the box. Why can the boat no longer land directly opposite, and which heading now gives the least drift?
Exam tip: resolve every velocity parallel and perpendicular to the bank, because the crossing time depends only on the perpendicular component v cosθ, while the drift depends only on the parallel component u − v sinθ.