What Are the Dimensions of A, B, and D in the Rocket Velocity Equation?

[BioCore]

Homework Statement

Consider a Rocket in space, far removed from external gravitational influences, and suppose the engine starts up. As long as the engines are running, its instantaneous velocity v is a function of its mass. V is given by:

V = A + B log(D/m)

A, B, D are physical quantities, find the dimensions of A, B and D.

Homework Equations

V = A + B log(D/m)

The Attempt at a Solution

From here I will refer to dimensions as such: [], ex. [V] dimensions of velocity.

[V] = L/T
thus [A] = L/T

I need to still find the and [D]. I am not entirely sure here of how to approach. But would the dimensions of log be nothing, in other words [D] = M or mass.

and thus = L/T as well. Or it this a stupid assumption to make?

Thank you for the help.

 

[Doc Al]

Makes sense to me.

 

[Moetasim]

I would approach this problem by first identifying the units of each term in the equation. The left side of the equation, V, represents velocity and has units of length per time ([V] = L/T). The first term on the right side, A, also represents velocity and has the same units as V ([A] = L/T).

Now, let's look at the second term, B log(D/m). The log function is dimensionless, so we can ignore it for now. The remaining terms, B and D/m, must have the same units in order for the equation to be dimensionally consistent.

Let's start with B. We know that must have units of length per time, so let's look at the units of D/m. D has units of length ([D] = L) and m has units of mass ([m] = M). So, D/m must have units of length per mass ([D/m] = L/M). Therefore, must also have units of length per mass ( = L/M).

Now, let's look at D. We know that [D] = L, but we also know that D represents a physical quantity. In this case, it represents a distance (such as the diameter of the rocket). Therefore, we can say that [D] = L (length) or [D] = [L] (distance).

In summary, the dimensions of A, B, and D are:
[A] = L/T (length per time)
= L/M (length per mass)
[D] = L or [D] = [L] (length or distance)

Note that the dimensions of D can be written as simply L, since distance is a fundamental dimension and does not require a unit.

I hope this helps! Keep up the good work in your studies.