In the context of fundamental quantities, mass can be expressed in terms of velocity (V), time (T), and force (F). We can use dimensional analysis to determine the dimensions of mass.
Let's consider the formula for force:
F = ma
Where: F represents force, m represents mass, and a represents acceleration.
The dimensions of force can be expressed as:
[F] = [M][LT^-2]
Where: [M] represents the dimension of mass, and [L] and [T] represent the dimensions of length and time, respectively.
Now, let's examine the formula for acceleration:
a = ΔV/ΔT
Where: ΔV represents change in velocity, and ΔT represents change in time.
Rearranging the equation, we get:
a = V/T
Rearranging further, we can express velocity as:
V = aT
Now, substituting this expression for velocity in terms of acceleration and time, into the dimensional formula for force, we have:
[F] = [M][LT^-2] = [M][(LT^-1)(T^-1)]
Comparing the dimensions, we can equate the exponents on both sides of the equation:
1 = -1 + (-1) + [M]
Simplifying, we get:
0 = [M]
Therefore, the dimension of mass ([M]) is dimensionless when considering velocity, time, and force as fundamental quantities.