posted by desperate help on .
A model rocket is launched straight upward with an initial speed of 30.0 m/s. It accelerates with a constant upward acceleration of 2.50 m/s2 until its engines stop at an altitude of 170 m.
(a) What is the maximum height reached by the rocket?
(b) How long after lift-off does the rocket reach its maximum height?
(c) How long is the rocket in the air?
Maximum height occurs where the velocity is temporarily zero.
While accelerating, the height is
y(t) = 30 t + 1.25 t^2
and the speed is
v(t) = 30 + 2.5 t
First figure out when y = 170m, and what the speed is at that time.
1.25 t^2 +30t -170 = 0
t^2 + 24t -136 = 0
That does not factor easily, so use the quadratic formula.
t = (1/2)(-24 + 33.47) = 4.73 s
v(t=4.73) = 30 + 11.83 = 41.83 m/s
Zero velocity will be reached after an additional time t', such that
g*t' = 41.83 m/s
t' = 4.27 s.
The total time after liftoff is then 9.00 seconds
The maximum altitude reached is
170 m + 41.83 t' -(g/2)t'^2
= 170 + 178.6 - 89.3 = 259.3 m
(b) t + t' = 9.00 s
(c) 9.00 s + (time to fall from 259.3 m)