Asked by Vijay Jaimini | 22nd Apr, 2012, 03:44: PM

Expert Answer:

This is a more common type of motion compared to the very restricted class of motions along a straight line. If you drive a car you turn left, right, and you go uphill and downhill even if this up and down motion is difficult to notice. When traveling by air three dimensions of motion are even more clearly pronounced.

For simplicity of representing the motion on graphs we will mostly discuss examples of two dimensional motion, but remember, adding movement in a third dimension does not change the basis of the theoretical description presented in this tutorial. Whenever we talk about motion in two dimensions the statements are also true for motion in three dimensions and vice versa. 

There is one fundamental law concerning motion in two or three dimensions – it can be decomposed into three independent motions each along one direction. A very classic example of this statement will be described later – it is projectile motion.

Now consider a quite common example – a person crossing a river in a boat. This motion is schematically shown in Fig. M2.1

 

 

 

 

 

Fig.M2.1

 

The boat starts perpendicularly to the bank line and so is the direction of the line drawn along the boat. The person on the boat has the feeling that he travels perpendicularly to the river bank, but an observer on solid ground sees the boat moving along a line that is tilted away from the perpendicular one.

The truth is that boat is moving independently and simultaneously into two directions: perpendicularly to the bank line – the green arrow indicates its velocity in this direction

and parallel to the bank line, with the river stream – as indicated by a dark blue arrow representing the velocity in this direction.  

The resulting velocity is indicated by the light blue arrow along the red arrow indicating the resulting direction of motion. The two velocities “green” and “blue” are independent. The velocity of the river current does not depend on the velocity of boat – that is obvious. The velocity of the boat, relative to the water,  does not depend on the velocity of the river current. We are not talking about mountain rivers, where the current may be so fast and turbulent that it prevents the motion of a boat powered by a small engine.

You can imagine moving very slowly across a very wide river, so that before reaching the opposite bank the boat will have moved downwards a few centimeters. This will be a three dimensional motion and motion in the third direction is also independent from the motions in the two other directions.

You can find many other examples of motions of an object with the resulting displacement being the sum of displacements in two or tree dimensions.

 

Motion of an object in one direction is independent of motion of the same object in another direction or directions.  

 

This is true if we do not consider the influence of forces on motion, that is if we study Kinematics, and for velocities much smaller then the velocity of light. It was already mentioned that velocity or speed of the order of 1000 km/s (kilometers per second!) may be considered as negligible compared to the speed of light, therefore all examples of motion we will describe in this part of the tutorial fulfill this criteria of “much smaller then the velocity of light.”

 

The independence of motion in different directions is the basis for analysis of all examples of motion in this paragraph.

 

The independence we are talking about holds not only for motions in different directions. Think about traveling by train or tram. If you walk around the carriage this motion is independent from the motion of the wagon itself even if you move in the same direction as a wagon. We exclude the effect of the shaking of the carriage, which may cause some difficulties in walking.

It is easy to imagine simultaneous motion of an object in more than three directions if you combine the motion of a person in the train with the motion of the Earth about the axis and around the Sun.

For experimental purposes four moving platforms can be constructed, each one smaller then the previous, each radio-controlled. They can be put one on another and each moved in slightly different directions.

The platform on top will experience simultaneous motion in four directions. For a well leveled  platform though, the motion will only be in two dimensions.

 

Do not confuse direction with dimension.

 

There is indefinite number of directions the object can move along, but there are only three independent dimensions in space.

 

What does independent dimension mean? In the Cartesian coordinate system the directions of x, y and z axis are independent. If the motion of an object is along the xaxis it is not possible to create such a motion by any combinations of motions along the y and z axis. The same is true for motion along any of the axis – it cannot be replaced by any combination of motion along two other axis.

Let’s consider an example of motion in two dimensions – it is easier to make the drawing for a such case.

The straight line motion in an arbitrary direction (except the previously described motion along the x or y axes) can be decomposed into simultaneous motions in xand y directions. This situation is depicted in Fig. M2.2.

 

 

 

 

Fig. M2.2

Motion of objects in different directions decomposed into motions in two independent directions x and y. These axes define a two dimensional space (in common language – simply a plane). Small red circles represent moving objects, red arrows their velocities. For objects denoted A and B those “red” velocities can be decomposed into two independent velocities along x and y axes. Object C is moving parallel to the x axis, therefore its velocity cannot be decomposed into any other direction. Or, formally you can say that y component of its velocity is zero, vyC=0.

is moving along y axis, so the x component of its velocity is zero, vxD=0.

Answered by  | 23rd Apr, 2012, 12:24: PM

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