A rocket, initially at rest on the ground, accelerates straight upward from rest with constant acceleration 29.4 . The acceleration period lasts for time 9.00 until the fuel is exhausted. After that, the rocket is in free fall. Find the maximum height reached by the rocket. Ignore air resistance and assume a constant acceleration due to gravity equal to 9.80 .

Well, looks like the rocket had a "blast"! Let's calculate the maximum height it reached.

During the acceleration phase, the rocket's acceleration is given as 29.4 m/s² and the time it takes to exhaust the fuel is 9.00 s. We can use the kinematic equation for displacement to find the velocity at the end of this phase:

v = u + at

Where:
v = final velocity
u = initial velocity (0 m/s since the rocket starts at rest)
a = acceleration (29.4 m/s²)
t = time (9.00 s)

Plugging in the values, we get:

v = 0 + 29.4 * 9.00
v = 264.6 m/s

Now, during the freefall phase, the rocket is subject to gravity alone, which provides an acceleration of 9.80 m/s². We can use the same kinematic equation to find the maximum height:

s = ut + (1/2)at²

Where:
s = displacement (maximum height)
u = initial velocity (264.6 m/s from the previous calculation)
a = acceleration due to gravity (-9.80 m/s² since the rocket is moving upward)
t = time (which we need to find)

Since the rocket is at its maximum height, its final velocity will be 0 m/s. Plugging in the values, we get:

0 = 264.6 * t + (1/2) * (-9.80) * t²
0 = 264.6t - (4.90t²)

Simplifying the equation:

4.90t² - 264.6t = 0

Now we can solve for t by factoring out a common term:

t(4.9t - 264.6) = 0

So either t = 0 (which we can ignore since we're looking for the maximum height) or:

4.9t - 264.6 = 0

Solving for t:

4.9t = 264.6
t = 264.6 / 4.9
t ≈ 54.08 s

Now that we have the time, we can find the maximum height by plugging it back into the displacement equation:

s = ut + (1/2)at²
s = 264.6 * 54.08 + (1/2) * (-9.80) * (54.08)²
s ≈ 7217.23 m

So, the maximum height reached by the rocket is approximately 7217.23 meters. That's quite an "uplifting" achievement!

To find the maximum height reached by the rocket, we can break the problem into two parts: the acceleration phase and the free-fall phase.

1. Acceleration Phase:
During the acceleration phase, the rocket accelerates upwards with a constant acceleration of 29.4 m/s² for a time of 9.00 s. To find the velocity at the end of this phase, we use the kinematic equation:

v = u + at

where:
v = final velocity
u = initial velocity (which is 0 m/s as the rocket starts from rest)
a = acceleration
t = time

v = 0 + 29.4 * 9.00
v = 264.6 m/s

Next, we can find the distance covered during the acceleration phase using the kinematic equation:

s = ut + 0.5at²

where:
s = distance
u = initial velocity
t = time
a = acceleration

s = 0 * 9.00 + 0.5 * 29.4 * (9.00)²
s = 1322.2 m

2. Free-Fall Phase:
After the fuel is exhausted, the rocket enters the free-fall phase. In this phase, the rocket is subject to the constant acceleration due to gravity, which is 9.80 m/s². To find the maximum height reached during free fall, we can use the kinematic equation:

v² = u² + 2as

where:
v = final velocity (which is 0 m/s at the highest point of the trajectory)
u = initial velocity
a = acceleration due to gravity
s = maximum height reached

By rearranging the equation, we can solve for the maximum height:

s = (v² - u²) / (2a)
s = (0 - 264.6²) / (2 * -9.80)
s = 1786.796 m

Therefore, the maximum height reached by the rocket is approximately 1786.796 meters.

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

Step 1: Find the time it takes for the rocket to reach its maximum height during the acceleration period.

In this problem, the rocket starts from rest, so its initial velocity (u) is 0. The acceleration (a) during the acceleration phase is given as 29.4 m/s^2. We need to find the time (t_acc) it takes for the rocket to reach its maximum height during this period.

We can use the kinematic equation: v = u + at, where v is the final velocity, u is the initial velocity, a is the acceleration, and t is the time.

Since the final velocity at the maximum height is 0 (as the rocket momentarily comes to rest before reversing direction), we have:

0 = 0 + (29.4 m/s^2) * t_acc

Simplifying the equation:

0 = 29.4 * t_acc

Since a product is equal to zero only if one or both of its factors is zero, we have two possible solutions for t_acc:

1) t_acc = 0 (initially at rest)
2) 29.4 * t_acc = 0 (final velocity at maximum height)

Since the second equation gives us the actual time during the acceleration phase, we discard the solution t_acc = 0.

Therefore, during the acceleration phase, it takes t_acc = 0 seconds for the rocket to reach its maximum height.

Step 2: Find the maximum height reached during the free fall period.

After the acceleration phase, the rocket is in free fall. During free fall, the only force acting on the rocket is gravity, and its acceleration is -9.80 m/s^2 (negative because it acts in the opposite direction to the rocket's motion).

To find the time it takes for the rocket to reach its maximum height during free fall, we can use the formula:

y = u*t + 0.5*a*t^2,

where y is the height, u is the initial velocity (0), a is the acceleration due to gravity (-9.80 m/s^2), and t is the time.

Since the rocket has come to rest at its maximum height, the final velocity (v) is 0.

0 = (0) * t_free + 0.5 * (-9.80 m/s^2) * t_free^2

Simplifying the equation:

0 = -4.90 * t_free^2

Since t_free^2 cannot be negative, we discard the negative solutions.

Therefore, during the free fall period, it takes t_free = 0 seconds for the rocket to reach its maximum height.

Step 3: Calculate the maximum height reached by the rocket.

During both the acceleration phase and the free fall period, the time taken to reach the maximum height for the rocket is 0 seconds. Therefore, the rocket reaches its maximum height instantaneously.

The maximum height reached by the rocket can be found by substituting the time t = 0 into the equation for height during free fall:

y = (0) * t + 0.5 * (-9.80 m/s^2) * t^2

Simplifying the equation:

y = 0

Thus, the maximum height reached by the rocket is 0 meters.

height reached during acceleration first

h = (1/2) a t^2 = (1/2)(29.4)(81) = 1191 meters
now speed up at 1190 meters
Vi = at = 29.4*9 = 265 m/s
so
we have a rocket at 1190 meters going at 265 m/s and with acceleration g = -9.8 m/s^2
When will the speed up hit zero
0 = Vi -gt
t = 265/9.8 = 27 seconds more up
how high then
h = Hi + Vi t + (1/2) a t^2
h = 1190 + 265*27 -4.9*27^2
h = 4773 m