A physical quantity is a phyical property of a phenomenon , body, or substance , that can be quantified by measurement.
The magnitude of the components of a vector are to be considered dimensionally distinct. For example , rather than an undifferentiated length unit L, we may represent length in the x direction as `L_(x)`, and so forth. This requirement status ultimately from the requirement that each component of a physically meaningful equation (scaler or vector) must be dimensionally consistent . As as example , suppose we wish to calculate the drift S of a swimmer crossing a river flowing with velocity `V_(x)` and of widht D and he is swimming in direction perpendicular to the river flow with velocity `V_(y)` relation to river, assuming no use of directed lengths, the quantities of interest are then `V_(x),V_(y)` both dimensioned as `(L)/(T)` , S the drift and D width of river both having dimension L. with these four quantities, we may conclude tha the equation for the drift S may be written : `S prop V_(x)^(a)V_(y)^(b)D^(c)`
Or dimensionally `L=((L)/(T))^(a+b)xx(L)^(c)` from which we may deduce that a+b+c=1 and a+b=0, which leaves one of these exponents undetermined. If, however, we use directed length dimensions, then `V_(x)` will be dimensioned as `(L_(x))/(T), V_(y) as (L_(y))/(T), S as L_(x)" and " D as L_(y)`. The dimensional equation becomes : `L_(x)=((L_(x))/(T))^(a) ((L_(y))/(T))^(b)(L_(y))^(c)` and we may solve completely as a=1,b=-1 and c=1. The increase in deductive power gained by the use of directed length dimensions is apparent.
From the concept of directed dimension what is the formula for a range (R) of a cannon ball when it is fired with vertical velocity component `V_(y)` and a horizontal velocity component `V_(x)`, assuming it is fired on a flat surface. [Range also depends upon acceleration due to gravity , g and k is numerical constant]
A. `R=(k(V_(x)V_(y)))/(g)`
B. `R=(k(V_(x))^(2))/(g)`
C. `R=(k(V_(x))^(3))/(V_(y)g)`
D. `R=(k(V_(y))^(3))/(V_(x)g)`
