1Which of these has the same dimensions as escape velocity?
Escape velocity and root mean square velocity both have the dimensions [L T⁻¹].
GravitationClass 11, Gravitation
Dimensional formula of Escape velocity
[L T-1]
The speed needed to leave a planet's gravity, sqrt(2GM/R). The 2 is invisible to dimensional analysis.
How to get the dimensional formula of escape velocity
- Start from the definition: escape velocity = sqrt(gravitational constant × mass / length).
- velocity = length / time, so velocity is [L] / [T] = [L T-1].
- acceleration = velocity / time, so acceleration is [L T-1] / [T] = [L T-2].
- force = mass × acceleration, so force is [M] × [L T-2] = [M L T-2].
- gravitational constant = force × length2 / mass2, so gravitational constant is [M L T-2] × [L]2 / [M]2 = [M-1 L3 T-2].
- Put in the dimension of each part: sqrt([M-1 L3 T-2] × [M] / [L]).
- Add the powers of each base quantity: [L T-1].
Quantities with the dimensions [L T⁻¹]
Equal dimensions do not make them the same quantity. Compare escape velocity with the others in the full grid.
Three questions on escape velocity
2What is the SI unit of escape velocity?
The SI unit of escape velocity is the metre per second (m/s).
3Which one has different dimensions from the others?
Gravitational potential is [L² T⁻²]. The other three are all [L T⁻¹].
Escape velocity: quick answers
What is the dimensional formula of escape velocity?
The dimensional formula of escape velocity is [L T⁻¹]. It comes from escape velocity = sqrt(gravitational constant × mass / length).
What is the SI unit of escape velocity?
The SI unit of escape velocity is the metre per second (m/s).
Which quantities have the same dimensions as escape velocity?
Escape velocity shares [L T⁻¹] with velocity, root mean square velocity, speed of light. Equal dimensions do not make them the same quantity.