Q4. A man walks at 1.5 m/s relative to a moving walkway, which moves at 1 m/s relative to the ground. How long does he take to cover the walkway’s 50 m length (a) walking with it, (b) walking against it?
(a) Relative to the ground his speed is 1.5 + 1 = 2.5 m/s, so 50/2.5 = 20 s. (b) 1.5 − 1 = 0.5 m/s, so 50/0.5 = 100 s.
Q5. A swimmer can swim at 2 m/s in still water. A river 120 m wide flows at 1 m/s. Find the shortest crossing time and the drift, and the time to cross straight to the opposite point.
Shortest time: 120/2 = 60 s, drift 1 × 60 = 60 m. Straight across: sin θ = 1/2, θ = 30° upstream; speed across √(4 − 1) = √3 ≈ 1.73 m/s, so about 69 s.
Q6. To a man walking east at 4 km/h, rain appears to fall vertically. When he doubles his speed, it appears to fall at 45° to the vertical, coming from the east. Find the actual speed and direction of the rain.
Let the rain’s velocity be p east and q down. At 4 km/h: p − 4 = 0, so p = 4. At 8 km/h, its velocity relative to him is (p − 8) = −4 east, that is 4 km/h westwards, and q down; at 45° the downward and horizontal parts are equal, so q = 4. So the rain falls at √(4 + 4) = 4√2 ≈ 5.7 km/h, at 45° to the vertical, moving eastwards (coming from the west).