Mechanics (Physics) question PLEASE ANSWER!?

Question

the HARP (High Altitude Research Project) gun can fire an 84.00 kg projectile containing scientific instrument staright up to an altitude of 179 km. If we pretend there is no air resistance , what muzzle spees is required to attain this altitude (m/s)? How long does the projectile remain at a height in excess of 100 km, the height of interest for high-altitude research (s)?

Answer 1

under ideal conditions, the projectile velocity will follow
v(t)=v0-g*t
and v(t)=0 at apogee

also
d(t)=v0*t-.5*g*t^2

solve for t and v0

v0=g*t, so
179000=v0*t-.5v0*t
v0*t=2*179000
plug into d(t)
179000=2*179000-.5*g*t^2
t=sqrt(2*179000/g)

v0=2*179000/sqrt(2*179000/g)

the total flight time is twice the time to apogee. To get the time above 100km, solve the following for t>0,
100000=v0*t-.5*g*t^2
double the t and subtract from the total flight time
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