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Relative Velocity in One Dimension

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Why this chapter
Why this chapter exists

A passenger in a train, a police car on a highway, two stones in the air. In each case one body is watched from another that moves. The first chapter said that rest and motion depend on the frame. This chapter puts numbers to that idea. One subtraction, done with signs, gives the velocity of any body as seen from any other. It also turns hard problems into easy ones, because you may sit in one of the moving bodies.

You will learn
By the end of this chapter you will
  1. Measure the position and velocity of one body relative to another on a line, and use the signs correctly.
  2. Change to the frame of a moving body by subtracting its velocity from every velocity, including the ground's.
  3. Solve chases, head-on approaches, and problems where a train passes a pole, a bridge or another train.
  4. Explain why two bodies in free fall move steadily relative to each other, and find where they meet.
  5. Read relative position and relative velocity from the x–t graphs of two bodies.
Before you begin
Before you begin

You should know Motion in One Dimension well. That means position and velocity with a sign convention, the three equations of motion, and free fall with upward positive. You should be able to convert km/h into m/s.

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Lessons

1
One Motion, Two Frames
The platform and the train from the first chapter, now with numbers. The same walk has one velocity for the passenger in the train and another for the man on the platform. The position of one body measured from another.
Same motion two framesRelative position on a line
13 min0%
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2
Relative Velocity on a Line
The rate at which the relative position changes. One subtraction, \(v_{AB} = v_A - v_B\), done with signs. The trick of sitting in one body and watching the others. The relative acceleration, which decides whether the relative motion is uniform.
Relative velocity on a lineSeeing the motion from a moving bodyRelative acceleration
15 min0%
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3
Chasing and Meeting
The two standard scenes of relative motion on a road or a track. A chase in the same direction, where the gap closes at the difference of the speeds. An approach from opposite directions, where the gap closes at their sum.
Bodies moving in the same directionBodies moving in opposite directions
14 min0%
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4
Crossing Lengths
When the length of a body matters. A train passing a pole, crossing a bridge or a platform, and passing another train. Each is solved by following one point of the train, with the relative velocity.
A train passing a pole or a bridgeTwo trains crossing each other
13 min0%
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5
Free Fall Together, and Graphs of Relative Motion
Two bodies thrown vertically share the same acceleration g, so gravity drops out of their relative motion and they move at a steady relative velocity. Then the same ideas read from x–t graphs, where the gap between two lines is the relative position.
Relative motion under gravityGraphs of relative motion
15 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.

frame of reference
An origin, an axis and a clock from which positions and times are measured. A frame is usually fixed to a body, such as the platform or a train.
free fall
Motion under gravity alone, with air resistance neglected. Every body in free fall near the Earth has the same acceleration \(g\), downward.
ground frame
The frame of reference fixed to the Earth. The platform, the road and the railway track are at rest in it. Velocities are measured in it unless a problem says otherwise.
observer
The person, real or imagined, who measures positions and times from a given frame of reference.
relative acceleration
The acceleration of one body as measured from another, \(a_{AB} = a_A - a_B\). It is zero when the two bodies have the same acceleration.
relative position
The position of one body measured from another. On a line, the position of A relative to B is \(x_{AB} = x_A - x_B\).
relative speed
The magnitude of the relative velocity, the rate at which the distance between two bodies changes. Same direction: the difference of the speeds. Opposite directions: their sum.
relative velocity
The velocity of one body as measured by an observer on another. On a line, the velocity of A relative to B is \(v_{AB} = v_A - v_B\), both measured in the same frame.