River Crossing and Swimmer-Boat Problems
The bank-frame velocity is v_ground = v_relative_water + v_river. For heading normal to the bank, crossing time is width divided by the cross-stream component; for zero drift, the along-stream components cancel.
Why this shows up in the exam
Planning ferry headings · Estimating drift in currents · Comparing fastest and shortest river routes
Learn the idea
Ground velocity across a river is the vector sum of velocity through water and river flow. The swimmer or boat controls velocity relative to water, while the river carries both downstream. Fastest crossing and shortest path are different optimizations.
🧠 Memory hook: Across sets the time; along sets the drift.
Get this one clearly and it pays off every single time it shows up in the paper. 🎯
Formulas & facts to keep ready
- v_ground = v_swimmer/water + v_water/ground — velocity addition between water and bank frames
- t_cross = W/|v_cross| — crossing time from cross-stream component
- v_swimmer,along + v_river = 0 — condition for reaching the directly opposite point
How to approach it
- 1Draw flow, across-river, and heading arrows
- 2Resolve the controlled velocity
- 3Use cross component for time and stream component for drift
Common slip-ups that cost marks
- •Using swimmer speed as ground speed
- •Confusing minimum time with minimum distance
- •Cancelling the wrong component for straight crossing
🌟 That's the whole idea — you've got this. Try the practice set below; every question you attempt makes it stick a little harder.
Original chapter practice
Original questions for this chapter, not past-paper questions or an exact mapping to this individual concept.
A particle has initial speed 2 m/s and constant acceleration 3 m/s^2 for 4 s. What distance does it cover?
More from Kinematics
Distance, Displacement, Speed, and Velocity
Understand the differences between distance and displacement, and between speed and velocity, including how to calculate average speed and average velocity in various scenarios.
Equations of Motion and Uniform Acceleration
Learn the kinematic equations for uniformly accelerated motion, including applications to free fall, retardation, and calculation of distance in specific time intervals.
Projectile Motion
Study the motion of projectiles, including the independence of horizontal and vertical components, trajectory equations, maximum height, and the effect of initial conditions.
Relative Velocity and Motion Analysis
Explore how to determine the velocity of one object relative to another and analyze motion from different reference frames, including periodic motion.
Position, Path Length, and Displacement
For one-dimensional motion, displacement over an interval is Delta x = x_f - x_i and may be positive, negative, or zero. Distance is the non-negative path length, so distance is always at least |Delta x|.
Speed and Velocity
Instantaneous velocity is the signed rate v = dx/dt, while instantaneous speed is |v|. In one dimension the sign of v identifies motion along or opposite the positive axis.