Answer:
The unit energy losses due to nonconservative forces is 881.40 joules per kilogram.
Explanation:
We can estimate the unit energy losses of gas eruption by Principle of Energy Conservation and Work-Energy Theorem:
[tex]U_{g,1} + K_{1} = U_{g,2}+K_{2}+W_{loss}[/tex] (Eq. 1)
Where:
[tex]U_{g,1}[/tex] - Gravitational potential energy of gas eruptions at surface, measured in joules.
[tex]U_{g,2}[/tex] - Gravitational potential energy of gas eruptions at highest height, measured in joules.
[tex]K_{1}[/tex] - Translational kinetic energy of gas eruptions at surface, measured in joules.
[tex]K_{2}[/tex] - Translational kinetic energy of gas eruptions at highest height, measured in joules.
[tex]W_{loss}[/tex] - Energy losses due to nonconservative forces, measured in joules.
We clear the component associated with energy losses in (Eq. 1):
[tex]W_{loss} = U_{g,1}-U_{g,2}+ K_{1}-K_{2}[/tex]
And we expand it afterwards:
[tex]W_{loss} = m\cdot g\cdot (z_{1}-z_{2}) + \frac{1}{2}\cdot m \cdot (v_{1}^{2}-v_{2}^{2})[/tex] (Eq. 2a)
[tex]w_{loss} = g\cdot (z_{1}-z_{2})+\frac{1}{2}\cdot (v_{1}^{2}-v_{2}^{2})[/tex] (Eq. 2b)
Where:
[tex]W_{loss}[/tex] - Energy losses due to nonconservative forces, measured in joules.
[tex]w_{loss}[/tex] - Unit energy losses due to nonconservative forces, measured in joules per kilogram.
[tex]g[/tex] - Gravitational acceleration, measured in meters per second.
[tex]z_{1}[/tex], [tex]z_{2}[/tex] - Bottom and top height, measured in meters.
[tex]v_{1}[/tex], [tex]v_{2}[/tex] - Gas eruption speeds at surface and highest heights, measured in meters per second.
If we know that [tex]g = 3.7\,\frac{m}{s^{2}}[/tex], [tex]z_{1} = 0\,m[/tex]. [tex]z_{2} =62\,m[/tex]. [tex]v_{1} = 36.111\,\frac{m}{s}[/tex] and [tex]v_{2} = 0\,\frac{m}{s}[/tex], the unit energy losses are:
[tex]w_{loss} = \left(3.7\,\frac{m}{s^{2}} \right)\cdot (62\,m-0\,m)+\frac{1}{2} \cdot \left[\left(36.11\,\frac{m}{s} \right)^{2}-\left(0\,\frac{m}{s} \right)^{2}\right][/tex]
[tex]w_{loss} = 881.40\,\frac{J}{kg}[/tex]
The unit energy losses due to nonconservative forces is 881.40 joules per kilogram.
