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Relative Velocity in Two Dimensions

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Why this chapter
Why this chapter exists

A boatman on the Ganga points his bow upstream to land at the opposite ghat. A runner tilts an umbrella forward in rain that falls straight down. A pilot flies a straight course with the nose pointing to one side. Each of them is working with a velocity measured from something that moves: the water, the runner, the air. This chapter gives you one rule for all of them, a subtraction of two vectors. It also gives you one picture: stand on one body and watch the other move in a straight line. With that picture, the least distance between two moving bodies becomes a perpendicular and a division.

You will learn
By the end of this chapter you will
  1. Find the velocity of one body relative to another by subtracting their velocities, tip to tail or in components.
  2. Solve river-boat problems: the shortest time, the shortest path, the drift, and the least drift when the river is faster than the boat.
  3. Find the apparent velocity of rain for a moving person, with and without wind, and the tilt of the umbrella.
  4. Find the heading and ground speed of an aeroplane in a wind, and compare round trips with and without wind.
  5. Find the least distance between two bodies moving at constant velocities and the time at which it occurs.
Before you begin
Before you begin

You should be able to add and subtract vectors, split a vector into components, and use sine, cosine and Pythagoras in a right triangle. You should know velocity as a vector from Motion in Two Dimensions, and the independence of perpendicular motions. The idea of a frame of reference from Motion in One Dimension helps.

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Lessons

1
Relative Velocity as a Vector
The velocity of one body as seen from another, found by subtracting two velocity vectors tip to tail. From the second body, the first moves in a straight line.
Relative velocity in a planeSubtracting velocities tip to tailThe frame of the other body
14 min0%
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2
The River and the Boat
A boat crossing a flowing river. Its velocity over the ground is its velocity through the water plus the velocity of the water. Crossing in the shortest time, crossing by the shortest path, and what to do when the river is faster than the boat.
Boat velocity relative to the groundCrossing in the shortest timeCrossing by the shortest pathWhen the river is faster than the boat
16 min0%
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3
Rain and the Umbrella
Rain as seen by a person who walks, runs or cycles. The rain relative to the person is the rain's velocity minus the person's velocity, and the umbrella must point along it. Rain driven by wind, and finding the true rain velocity from two observations.
Rain seen by a moving personRain with windFinding the true rain velocity
14 min0%
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4
The Aeroplane and the Wind
An aeroplane flies through air that itself moves. Its velocity over the ground is its velocity through the air plus the wind. Heading into the wind to hold a course, and why a wind always lengthens a round trip.
Airspeed heading and ground velocityFlying to a destination in a windA round trip in a wind
13 min0%
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5
Two Bodies and the Closest Approach
Two bodies moving in a plane at constant velocities. Seen from one of them, the other moves in a straight line. The least distance is then a perpendicular, and its time is a division. The condition for a collision.
Relative motion of two bodies in a planeMinimum distance between two bodiesTime of closest approachCondition for collision
16 min0%
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Read in order. The small labels are the topics of each lesson. They take your colour as you answer the checks.

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The Workbook
Questions to solve on paper, marked by you.
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Glossary

The key words of this chapter, with the meaning this chapter gives them.

airspeed
The speed of an aeroplane relative to the air around it.
apparent velocity
The velocity of a body as it seems to a moving observer, its velocity relative to that observer. For rain it is \(\vec v_r - \vec v_m\).
closest approach
The moment and the position at which two moving bodies are nearest to each other.
collision course
The motion of two bodies whose relative velocity points straight from one to the other, so that their separation falls to zero. Each sees the other at a constant bearing.
drift
The distance along the bank between the point exactly opposite the start and the point where a boat lands.
frame of reference
A body, or a set of axes fixed to it, from which positions and velocities are measured. The ground, a moving bus and a flowing river are all frames of reference.
ground speed
The speed of an aeroplane or a boat relative to the ground.
heading
The direction in which a boat or an aeroplane points, the direction of its velocity relative to the water or the air.
independence of motions
The principle that the motions of a body along two perpendicular axes do not affect each other. Each is solved on its own, and only the time is shared.
minimum distance
The least separation of two moving bodies. For constant velocities it is the perpendicular distance from one body to the straight path of the other in the first body's frame.
relative position
The position of one body measured from another, the vector drawn from the second body to the first: \(\vec r_{AB} = \vec r_A - \vec r_B\).
relative velocity
The velocity of a body as measured by an observer moving with another body: \(\vec v_{AB} = \vec v_A - \vec v_B\).
resultant
The single vector equal to the sum of two or more vectors. It closes the triangle or polygon formed by the vectors drawn tip to tail.
tip to tail
The rule for adding vectors: draw the second vector with its tail at the tip of the first. The sum runs from the tail of the first to the tip of the second.
track
The path of a boat or an aeroplane over the ground, the direction of its velocity relative to the ground. In a current or a wind the track differs from the heading.
wind velocity
The velocity of the air relative to the ground.