help me w/ this physic problem plz..?

Question

A model rocket is launched straight upward
with an initial speed of 47.8 m/s. It acceler-
ates with a constant upward acceleration of
2.19 m/s2 until its engines stop at an altitude
of 180 m.
a) What is the maximum height reached by
the rocket? Answer in units of m.

b) When does the rocket reach maximum
height? Answer in units of s.

c) How long is the rocket in the air? Answer
in units of s.

Answer 1

Ignoring air resistance and loss of mass during the burn. g=9.81 m/s^2
y(t)=180m=
47.8*t+.5*2.19*t^2

solve for t. This is the point at which the rocket no longer has engine power and gravity takes over the motion
t=3.487
plug that t into
v(t)=47.8+2.19*t

now you have the initial velocity at 180m, v0.
v0=55.4 m/s
Now the equations of motion change to
y(t)=180+v0*t-.5*g*t^2
and
v(t)=v0-g*t

to find max height, set v(t)=0 and solve for t. Plug that t into
y(t).

To find when the rocket hits the ground, set y(t)=0 and solve for the positive root.

The apogee occurs at
5.65 seconds after burn

Max altitude is
336.6 meters

The return to Earth occurs at
13.935 after burn

total flight time is
13.935+3.487 seconds


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