What Quantities Determine the Velocity of a Cover Thrown by an Explosion?

[evinda]

Hello! (Wave)

A thin cover with the shape of a rectangle with mass per unit of volume equal to $m_f$ is put over a quantity of explosive ( with mass per unit of volume equal to $m_e$), that is attached at a base of a practically unbounded mass. If the explosive explodes, the cover is getting thrown vertically with velocity $v_f$. If $E_g$ is the so called "energy Gurney" of the explosive (in unites Joules/kg), i.e. the energy that the explosive material disposes to produce work, determine, as accurate as possible, using techniques of dimensional analysis, the velocity of the cover as a fuction of $m_f, m_e$ and $E_g$. (The exact relation that holds is the following: $v_f=\sqrt{2E_g} \left(\frac{m_f}{m_e}+\frac{1}{3} \right)^{-\frac{1}{2}}$)Could you help me to find the quantities that we will use? (Thinking)

 

[jvicens]

Hello! I would first start by breaking down the problem into its basic units. We have mass per unit volume, which is typically measured in kilograms per cubic meter (kg/m^3). We also have velocity, which is measured in meters per second (m/s). Finally, we have energy, which is measured in Joules (J).

Using dimensional analysis, we can determine the units of our final answer. The velocity, $v_f$, will have units of m/s. The mass per unit volume of the cover, $m_f$, will have units of kg/m^3. The mass per unit volume of the explosive, $m_e$, will also have units of kg/m^3. And the energy Gurney, $E_g$, will have units of J/kg.

Now, we can use these units to create a dimensionally consistent equation. We know that velocity is equal to distance divided by time, so we can set up our equation as follows:

$v_f = \frac{E_g}{m_e/m_f} = \frac{E_gm_f}{m_e}$

Next, we can use the given relation to solve for the velocity of the cover:

$v_f = \sqrt{2E_g} \left(\frac{m_f}{m_e}+\frac{1}{3} \right)^{-\frac{1}{2}}$

By substituting our units into this equation, we can see that it is dimensionally consistent. Therefore, we can say with confidence that the velocity of the cover, $v_f$, is a function of the mass per unit volume of the cover, $m_f$, the mass per unit volume of the explosive, $m_e$, and the energy Gurney, $E_g$.

I hope this helps in finding the quantities needed to solve the problem. Good luck!