# 4.5 Relative motion in one and two dimensions

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By the end of this section, you will be able to:
• Explain the concept of reference frames.
• Write the position and velocity vector equations for relative motion.
• Draw the position and velocity vectors for relative motion.
• Analyze one-dimensional and two-dimensional relative motion problems using the position and velocity vector equations.

Motion does not happen in isolation. If you’re riding in a train moving at 10 m/s east, this velocity is measured relative to the ground on which you’re traveling. However, if another train passes you at 15 m/s east, your velocity relative to this other train is different from your velocity relative to the ground. Your velocity relative to the other train is 5 m/s west. To explore this idea further, we first need to establish some terminology.

## Reference frames

To discuss relative motion in one or more dimensions, we first introduce the concept of reference frame     s . When we say an object has a certain velocity, we must state it has a velocity with respect to a given reference frame. In most examples we have examined so far, this reference frame has been Earth. If you say a person is sitting in a train moving at 10 m/s east, then you imply the person on the train is moving relative to the surface of Earth at this velocity, and Earth is the reference frame. We can expand our view of the motion of the person on the train and say Earth is spinning in its orbit around the Sun, in which case the motion becomes more complicated. In this case, the solar system is the reference frame. In summary, all discussion of relative motion must define the reference frames involved. We now develop a method to refer to reference frames in relative motion.

## Relative motion in one dimension

We introduce relative motion in one dimension first, because the velocity vectors simplify to having only two possible directions. Take the example of the person sitting in a train moving east. If we choose east as the positive direction and Earth as the reference frame, then we can write the velocity of the train with respect to the Earth as ${\stackrel{\to }{v}}_{\text{TE}}=10\phantom{\rule{0.2em}{0ex}}\text{m}\text{/}\text{s}\phantom{\rule{0.2em}{0ex}}\stackrel{^}{i}$ east, where the subscripts TE refer to train and Earth. Let’s now say the person gets up out of /her seat and walks toward the back of the train at 2 m/s. This tells us she has a velocity relative to the reference frame of the train. Since the person is walking west, in the negative direction, we write her velocity with respect to the train as ${\stackrel{\to }{v}}_{\text{PT}}=-2\phantom{\rule{0.2em}{0ex}}\text{m}\text{/}\text{s}\phantom{\rule{0.2em}{0ex}}\stackrel{^}{i}\text{.}$ We can add the two velocity vectors to find the velocity of the person with respect to Earth. This relative velocity    is written as

${\stackrel{\to }{v}}_{\text{PE}}={\stackrel{\to }{v}}_{\text{PT}}+{\stackrel{\to }{v}}_{\text{TE}}.$

Note the ordering of the subscripts for the various reference frames in [link] . The subscripts for the coupling reference frame, which is the train, appear consecutively in the right-hand side of the equation. [link] shows the correct order of subscripts when forming the vector equation. When constructing the vector equation, the subscripts for the coupling reference frame appear consecutively on the inside. The subscripts on the left-hand side of the equation are the same as the two outside subscripts on the right-hand side of the equation.

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