Past exam of the physics 1 paper of the University of Oxford / 2019 resit / 1

The dimensions of mass density, time and energy are $\mathsf{M}{\mathsf{L}}^{-3}$, $\mathsf{T}$ and $\mathsf{M}{\mathsf{L}}^2{\mathsf{T}}^{-2}$. Matching powers in the proposed dimensional analysis givesThusThis is the Sedov-Taylor blast wave scaling for a fixed released energy. Dimensional analysis cannot determine the dimensionless $C$; that requires the fluid equations and thermodynamic assumptions. The assumed list of controlling parameters excludes, for example, an independent ambient sound speed or a continuing energy supply.