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**1. 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 dependance of its distance z from the center of the earth.

**2. 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 dont 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!