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Homework Statement
A mass m is thrown vertically from the surface of the Earth with a velocity v0. Find a function that describes the velocity v of the mass m in dependence of its distance z from the center of the earth.
Homework Equations
m*z'' = -G* m*M / z^2
z'' = dv / dt
G = gravitational constant
M = mass of earth
3. the solution and my problem with it
z'' = -G * M * 1/z^2
z'' = dv/dt = v * dv/dz
==> v * dv = -G * M * dz/z^2
Now my Problem: In the next part both sides of the above equation are to be integrated. The left side from v0 to v, and the right side from R to z. Like that:
integral(v0..v, v*dv) = -G * M * integral(R..z, dz/z^2)
Unfortunately, I don't understand that. Why can I inegrate both sides of the equation like that, and why is it still the same afterwards? What exactly does the term on the left 'integral(v0..v, v*dv)' and the term on the right '-G * M * integral(R..z, dz/z^2)' side of the above equation mean?
It would be great if someone could provide a little help!