1.17 Understanding motion  (Page 2/3)

For two or three dimensional motion,

For one dimensional motion,

Similarity / Difference 6 : We can not draw position – time, displacement – time or velocity – time plots for three dimensional motion. We can draw these plots for two dimensional motion, but the same would be complex and as such we would avoid drawing them.

We can, however, draw the same for one – dimensional motion by treating the vector attributes (position vector, displacement and velocity) as scalar with appropriate sign. Such drawing would be in the first and fourth quarters of two – dimensional plots. We should clearly understand that if we are drawing these plots, then the motion is either one or two dimensional. In general, we draw attribute .vs. time plots mostly for one – dimensional motion. We should also understand that graphical method is an additional tool for analysis in one dimensional motion.

The slope of the curve on these plots enables us to calculate the magnitude of higher attributes. The slope of position – time and displacement – time plot gives the magnitude of velocity; whereas the slope of velocity – time plot gives the magnitude of acceleration (This will be dealt in separate module).

Significantly, the tangent to the slopes on a "time" plot does not represent direction of motion. It is important to understand that the though the nature of slope (positive or negative) gives the direction of motion with respect to reference direction, but the tangent in itself does not indicate direction of motion. We must distinguish these “time” plots with simple position plots. The curve on the simple position plot is actual representation of the path of motion. Hence, tangent to the curve on position plot (plot on a x,y,z coordinate system) gives the direction of motion.

Similarity / Difference 7 : Needless to say that what is valid for one dimensional motion is also valid for the component motion in the case of two or three dimensional motion. This is actually a powerful technique to even treat a complex two or three dimensional motion, using one dimensional techniques. This aspect will be demonstrated on topics such as projectile and circular motion.

Similarity / Difference 8 : The area under velocity – time plot (for one dimensional motion) is equal to displacement.

As area represents a vector (displacement), we treat area as scalar with appropriate sign for one dimensional motion. The positive area above the time axis gives the positive displacement, whereas the negative area below time axis gives negative displacement. The algebraic sum with appropriate sign results in net displacement. The algebraic sum without sign results in net distance.

Important thing to realize is that this analysis tool is not available for analysis of three dimensional motion as we can not draw the plot in the first place.

Similarity / Difference 9 : There is a difficulty in giving differentiating symbols to speed and velocity in one dimensional motion as velocity is treated as scalar. Both are represented as simple letter “v”. Recall that a non-bold faced letter “v” represents speed in two/three dimensional case. An equivalent representation of speed, in general, is |v|, but is seldom used in practice. As we do not use vector for one dimensional motion, there is a conflicting representation of the same symbol, “v”. We are left with no other solution as to be elaborate and specific so that we are able to convey the meaning either directly or by context. Some conventions, in this regard, may be helpful :