A rocket accelerates straight up from rest for 20 seconds with an acceleration of 8 m/s2. After 20 seconds the rocket motor shuts off and the rocket continues to coast upwards. How high does the rocket go in meters?

During fuel burn,

v = at = 8*20 = 160

after that, the height

h = 160t - 4.9t^2
the vertex of the parabola occurs when t = 160/9.8 = 16.32 seconds

At that point, h(t) = 1306.12 m

To determine how high the rocket goes, we need to calculate the distance traveled during the acceleration phase and the distance traveled during the coasting phase.

During the acceleration phase, the rocket starts from rest (initial velocity, u = 0) and accelerates for 20 seconds with an acceleration (a) of 8 m/s². We can use the kinematic equation:

s = ut + (1/2)at²

where s is the distance traveled, u is the initial velocity, t is the time, and a is the acceleration.

For the acceleration phase:
u = 0 (since the rocket starts from rest)
a = 8 m/s²
t = 20 s

Plugging the values into the equation, we get:
s = 0(20) + (1/2)(8)(20²)
s = 0 + (4)(400)
s = 1600 m

Therefore, during the acceleration phase, the rocket travels 1600 meters.

After the motor shuts off, the rocket continues to move upward without any further acceleration. This is known as the coasting phase. During this phase, the rocket's velocity remains constant.

To determine the distance traveled during the coasting phase, we need to know the rocket's velocity at the end of the acceleration phase. We can calculate this by multiplying the acceleration by the time:

v = u + at

During the acceleration phase:
u = 0
a = 8 m/s²
t = 20 s

v = 0 + (8)(20)
v = 160 m/s

So, the rocket's velocity at the end of the acceleration phase is 160 m/s.

During the coasting phase, the rocket's velocity remains constant at 160 m/s for a certain amount of time. Let's call this time duration "t_coast".

During the coasting phase, the distance traveled (s_coast) is given by:
s_coast = v × t_coast

Since the rocket is coasting upward, the final displacement will be positive. Therefore, during the coasting phase, the distance traveled is:
s_coast = 160(t_coast)

We don't have information about the duration of the coasting phase, but we know that the total time is 20 seconds. Hence, the duration of the coasting phase will be:
t_coast = total time - acceleration time
t_coast = 20 - 20
t_coast = 0 s

Since the coasting phase has a duration of 0 seconds, the distance traveled during this phase is also 0 meters.

Therefore, the total distance traveled by the rocket is the sum of the distances traveled during the acceleration and coasting phases:

Total distance = Distance during acceleration + Distance during coasting
Total distance = 1600 + 0
Total distance = 1600 meters

So, the rocket goes up to a height of 1600 meters.