Question #6cea0

1 Answer
Jul 1, 2015

I think it's ["M"^0"L T"^(-1)].

Explanation:

Dimensional formulas use mass, length, and time to express physical quantities as a propduct of powers of fundamental quantities.

Since velocity is defined as distance divided by time, you can say that

"velocity" = "distance"/"time" = "meters"/"seconds" = "m"/"s" = "m s"^(-1)

This means that the dimensional formula for velocity will have

["M"^(0) "L T"^(-1)] <=> ["L T"^(-1)]

This tells you that

  • you have no units of mass in equation, since "M" is raised to the power of zero;
  • you have units of length, meters, raised to the power of 1;
  • you have units of time, seconds, raised to the power of negative 1;

So, take the equation for escape velocity and test to see if these are the units that you're left with for the dimensional formula.

v_"esc" = sqrt((2 * G * M)/r), where

G - the universal gravitational constant, equal to 6.67 * 10^(-11)"m"^(3)"kg"^(-1)"s"^(-2)
M - the mass of the body, expressed in "kg";
r - the distance from the center of the body's mass to a point in space.

If you use units, the equation will look like this

v_"esc" = sqrt( ((2 * "m"^(3)cancel("kg"^(-1))"s"^(-2) * cancel("kg"))/"m")

v_"esc" = sqrt( (2 * "m"^(2) * "s"^(-2))

As far as units are concerned, this will give you

v_"esc" = "m" * "s"^(-1)

As you can see, this confirms the initial dimensional formula, since you have meters multiplied by seconds to the power of negative 1. Therefore, the dimensional formula for escape velocity is

["M"^(0)"L T"^(-1)] <=> ["L T"^(-1)]