A rocket moves upward from rest with an acceleration of 29.4 m/s2 for 4 seconds. It then runs out of fuel and continues to move upward. Neglecting any air resistance, how high will it rise?

under power

h = 14.7*4^2 = 235.2
v = 29.4*4 = 117.6

on a ballistic path,
h = 235.2 + 117.6t - 4.9t^2
max height is at t = 117.6/9.8
plug that into h to find the height.

To find the height the rocket will rise, we need to calculate the distance covered during the period of acceleration and the distance covered during the period of deceleration.

Step 1: Calculate the distance covered during acceleration.
During the period of acceleration, the rocket's initial velocity is 0 m/s. We can use the kinematic equation:
\[ d = v_i t + \frac{1}{2} a t^2 \]
where:
d = distance
v_i = initial velocity
a = acceleration
t = time

Plugging in the values:
\[ d = 0 (4) + \frac{1}{2} (29.4) (4)^2 \]

Simplifying:
\[ d = \frac{1}{2} (29.4) (16) \]

Calculating:
\[ d = 235.2 \, \text{m} \]

So, during the period of acceleration, the rocket covers a distance of 235.2 m.

Step 2: Calculate the distance covered during deceleration.
After the rocket runs out of fuel, it continues to move upward due to the upward velocity it gained during the period of acceleration. Since the initial velocity is 0 m/s, the rocket will decelerate under the influence of gravity (-9.8 m/s^2).

To calculate the distance covered during deceleration, we can use the same kinematic equation mentioned earlier. However, this time the acceleration will be negative (-9.8 m/s^2).
Plugging in the values:
\[ d = 0 (t) + \frac{1}{2} (-9.8) t^2 \]

We need to find the time at which the rocket starts decelerating. This can be calculated by subtracting the initial acceleration period (4 seconds) from the total time (4 seconds).
\[ t = 4 - 4 \]

Simplifying:
\[ t = 0 \, \text{s} \]

Since the time is zero, the rocket won't cover any distance during deceleration.

Step 3: Calculate the total distance covered.
To find the total distance covered by the rocket, we add the distances covered during acceleration and deceleration:
\[ d_{\text{total}} = d_{\text{acceleration}} + d_{\text{deceleration}} \]
\[ d_{\text{total}} = 235.2 \, \text{m} + 0 \, \text{m} \]
\[ d_{\text{total}} = 235.2 \, \text{m} \]

Therefore, the rocket will rise to a height of 235.2 meters.

To determine the height the rocket will reach, we can break down the problem into two parts: the time when the rocket has fuel and the time after it runs out of fuel.

First, let's calculate the velocity of the rocket after 4 seconds when it still has fuel. We can use the equation:

v = u + at,

where:
v = final velocity (unknown),
u = initial velocity (which is 0 m/s since it starts from rest),
a = acceleration (given as 29.4 m/s^2), and
t = time (given as 4 seconds).

Plugging in the values, we have:

v = 0 + (29.4 m/s^2)(4 s) = 117.6 m/s.

Therefore, the velocity of the rocket after 4 seconds is 117.6 m/s.

Next, we can calculate the maximum height reached by the rocket. To do this, we need to find the time it takes for the rocket to reach the peak of its trajectory. This can be done using the equation:

v = u + at,

where now:
v = final velocity (which is 0 m/s at the peak),
u = initial velocity (117.6 m/s, as calculated above),
a = acceleration (which is the acceleration due to gravity, approximately 9.8 m/s^2), and
t = time (unknown).

Plugging in the values, we have:

0 = 117.6 m/s + (-9.8 m/s^2)(t).

Simplifying the equation, we get:

t = (117.6 m/s) / (9.8 m/s^2) ≈ 12 s.

Therefore, it takes approximately 12 seconds for the rocket to reach its peak.

Now we can calculate the maximum height reached by the rocket using the equation:

h = ut + 0.5at^2,

where:
h = height (unknown),
u = initial velocity (117.6 m/s),
t = time (12 seconds), and
a = acceleration (also negative since it opposes the rocket's motion, -9.8 m/s^2).

Plugging in these values, we have:

h = (117.6 m/s)(12 s) + 0.5(-9.8 m/s^2)(12 s)^2.

Simplifying the equation, we get:

h = 1411.2 m + (-70.56 m/s^2)(144 s^2) ≈ 1411.2 m - 10151.68 m = -8740.48 m.

Since height cannot be negative (assuming the ground is at y = 0), the rocket will not fall below the ground level. Therefore, the maximum height reached by the rocket is approximately 8740.48 meters.