A model rocket is launched straight upward with an initial speed of 50.2 m/s. It accelerates with a constant upward acceleration of 2.12 m/s2 until its engines stop at an altitude of 167 m. What is the maximum height reached by the rocket?

h = 50.2t + (2.12-9.8)/2 t^2

= 50.2t - 3.84t^2
engine kicks out when h=157
157 = 50.2t - 3.84t^2
t = 5.18

now the velocity is 50.2 + 5.18*2.12 = 61.18

now, from that point, reset the clock to 0, and its height is ballistic:

h = 157 + 61.18t - 4.9t^2
h = 157 + t(61.18 - 4.9t)
max height reached at t = 6.24
h(6.24) =~ 348m

time from blast-off is 5.18+6.24 = 11.82 seconds

You didn't do your work you put in a time how am I suppose to see it is correct without work to prove it. You go to math to learn proofs so use the skill if you are going to publish things. Don't be lazy

The engine doesn't kick out at 157, it kicks out at 167.

To find the maximum height reached by the rocket, we need to determine the time it takes for the engines to stop and then use that time to calculate the height.

First, let's find the time it takes for the engines to stop. We can use the equation of motion:

v = u + at

where:
v = final velocity of the rocket (0 m/s, since the engines stop)
u = initial velocity of the rocket (50.2 m/s)
a = acceleration of the rocket (2.12 m/s²)
t = time

Rearranging the equation, we have:

t = (v - u) / a

Plugging in the values:

t = (0 - 50.2) / 2.12
t = -50.2 / 2.12
t ≈ -23.68 seconds

Since time cannot be negative, we discard the negative sign and take the absolute value of t:

t ≈ 23.68 seconds

Now that we know the time it takes for the engines to stop, we can calculate the maximum height reached by the rocket using the kinematic equation:

s = ut + (1/2)at²

where:
s = displacement (height) of the rocket
u = initial velocity (50.2 m/s)
t = time (23.68 seconds)
a = acceleration (2.12 m/s²)

Plugging in the values:

s = (50.2 * 23.68) + (1/2)(2.12)(23.68)^2
s = 1187.44 + (1/2)(2.12)(561.5424)
s ≈ 1187.44 + 596.2407
s ≈ 1783.6807

Therefore, the maximum height reached by the rocket is approximately 1783.68 meters.