For the first minute, the rocket is traveling straight upward at a constant velocity of 600 m/s. This means that after 1 minute (60 seconds), it will have risen to an altitude of
[tex]y=\left(600\,\dfrac{\rm m}{\rm s}\right)(60\,\mathrm s)=36000\,\mathrm m[/tex]
From this point on, it's in freefall with a starting height of 36000 m, a starting velocity of 600 m/s upward, and accelerating downward with a magnitude of 1g. At its maximum height, the rocket's velocity will be 0. We have a formula that ties all this information together:
[tex]-\left(600\,\dfrac{\rm m}{\rm s}\right)^2=2(-g)(y_{\rm max}-36000\,\mathrm m)[/tex]
[tex]\implies y_{\rm max}=54367\,\mathrm m\approx54000\,\mathrm m[/tex]
(rounding to 2 significant digits)
The altitude [tex]y[/tex] of the rocket with respect to time [tex]t[/tex] (AFTER the first minute) is given by
[tex]y=36000\,\mathrm m+\left(600\,\dfrac{\rm m}{\rm s}\right)t-\dfrac g2t^2[/tex]
The time it takes for the rocket to reach the ground ([tex]y=0[/tex]) is given by
[tex]0=36000\,\mathrm m+\left(600\,\dfrac{\rm m}{\rm s}\right)t-\dfrac g2t^2[/tex]
[tex]\implies t=166.56\,\mathrm s[/tex]
which means the rocket will have spent a TOTAL time of 226.56 seconds, or approximately 230 seconds, in the air.
