*Notes to a video lecture on http://www.unizor.com*

__Relative Motion__

The subject of this topic is motion of one object relative to another. In particular, assume that we have an original

*inertial frame of reference*and two objects moving with constant velocities in this frame.

Consider further that we would like to use another frame of reference -

the one associated with one of the objects and axes parallel to the axes

of the original frame of reference.

As we have proven before, this new frame of reference is

*inertial*. Our task now is to determine the vectors of

*position*and

*velocity*of objects in this new frame of reference.

As a typical example, consider two trains, A and B, going in opposite

directions along two straight lines parallel to each other with constant

speeds.

Our original frame of reference is the ground. We choose an X-axis along

the train lines with positive direction coinciding with the direction

of train A. Then Y- and Z-coordinates of positions and velocities of

both trains are zero.

The task at hand is to determine how fast train B passes train A from the viewpoint of a passenger of train A.

Obviously, translated this task to a language of frames of reference, we

have to determine the characteristics of motion of the train B in an

inertial frame associated with train A.

Let

*and*

**P**_{a}(t)*be*

**P**_{b}(t)*position vectors*

of our two objects, A and B, in the original frame of reference. These

positions, obviously, depend on time since objects A and B are moving.

Their

*velocity vectors*are, correspondingly,

*and*

**V**_{a}*, which we assume are constant.*

**V**_{b}Vector

*represents the*

**P**_{ab}(t)*position vector*of object B in the new frame of reference associated with object A.

Since, according to the rules of vectors' addition,

*=*

**P**_{b}(t)*+*

**P**_{a}(t)

**P**_{ab}(t)we can find

*:*

**P**_{ab}(t)*=*

**P**_{ab}(t)*−*

**P**_{b}(t)

**P**_{a}(t)Differentiating this by time, we obtain the corresponding equations for

*velocity vectors*:

*=*

**V**_{b}*+*

**V**_{a}

**V**_{ab}*=*

**V**_{ab}*−*

**V**_{b}

**V**_{a}The equations above, for positions and velocities, determine the motion

of object B in an inertial frame of reference associated with object A

and express the

**Galileo's Relativity Principles**.

Let's apply this rule to a task with two trains, A and B, moving in opposite directions along the X-axis with speeds

**V**_{a}= 100 km/hour

**V**_{b}= −90 km/hour(minus for train B signifies that, if the positive direction of the

X-axis is towards the movement of train A, train B moves in an opposite

direction).

Then the relative speed of train B for a passenger on train A is

*=*

**V**_{ab}*−*

**V**_{b}*=*

**V**_{a}=

*−90 − 100 = −190*The minus sign means that a passenger on train A sees train B moving

backwards (relative to his own movement, which is towards positive

direction of the X-axis). The absolute value of a relative speed in this

case of motion in opposite directions, as we see, is a sum of absolute

values of corresponding speeds.

Let's assume that the trains move in the same (positive) direction of

the X-axis with above speeds. Now their velocity vectors are

**V**_{a}= 100 km/hour

**V**_{b}= 90 km/hourThen the relative speed of train B for a passenger on train A is

*=*

**V**_{ab}*−*

**V**_{b}*=*

**V**_{a}=

*90 − 100 = −10*The minus sign means that a passenger on train A sees train B moving

backwards (because train A is faster and, therefore, train B, relative

to a passenger on train B, moves backward). The absolute value of a

relative speed in this case of motion in the same direction, as we see,

is a difference of absolute values of corresponding speeds.

Other examples of velocities' arithmetic are as follows.

1. A platform A moves uniformly along straight railing with speed

**V**_{a}= 3 m/sec

**V**_{ab}= 1 m/sec

**V**_{b}Let's associate our original frame of reference with the ground with X-axis directed along the direction of a moving platform.

Since

*=*

**V**_{b}*+*

**V**_{a}

**V**_{ab}

**V**_{b}= 3 + 1 = 4 (m/sec)2. A river A flows uniformly along straight banks with speed

*.*

**V**_{a}= 3 m/sec*.*

**V**_{ab}= 10 m/sec

**V**_{b}Let's associate our original frame of reference with the ground with X-axis directed along the flow of the river.

Then

*.*

**V**_{a}= 3 m/secIf the boat goes down the river, its speed relative to water is

*.*

**V**_{ab}= 10 m/secIf the boat goes up the river, its speed relative to water is

*.*

**V**_{ab}= −10 m/secFrom the same equation for speeds

*=*

**V**_{b}*+*

**V**_{a}

**V**_{ab}follows that, when the boat goes down the river, its speed is

**V**_{b}= 3 + 10 = 13 (m/sec)and up the river

**V**_{b}= 3 − 10 = −7 (m/sec)(here minus means that the boat moves against the positive direction of the X-axis, associated with the flow of the river).

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