A rocket is launched with some initial velocity and travels with constant acceleration of 25.7 m/s2. When the rocket has reached a height of 227 m above its launch position, its velocity is 154 m/s. How long did it take the rocket to reach this height?




We can solve this problem using the **kinematic equations** for motion under constant acceleration. The kinematic equation that relates displacement, initial velocity, acceleration, and time is:

\[
v = u + at
\]

\[
s = ut + \frac{1}{2} a t^2
\]

Where:
- \(v\) is the final velocity (154 m/s),
- \(u\) is the initial velocity (unknown),
- \(a\) is the acceleration (25.7 m/s²),
- \(s\) is the displacement or height (227 m),
- \(t\) is the time (what we need to find).

### Step 1: Rearranging the Equation
First, we need to find the time taken. To do this, we'll start by using the following kinematic equation that involves displacement \(s\), initial velocity \(u\), acceleration \(a\), and time \(t\):

\[
s = ut + \frac{1}{2} a t^2
\]

This simplifies to:

\[
227 = u t + \frac{1}{2} (25.7) t^2
\]

This equation depends on the unknown initial velocity \(u\), so we’ll need a second equation to solve for \(u\) as well.

### Step 2: Use the Velocity Equation
We also know the velocity at the height of 227 m is 154 m/s. Using the equation:

\[
v = u + at
\]

Substitute the known values:

\[
154 = u + 25.7 t
\]

Now, we have a system of two equations:
1. \( 227 = u t + 12.85 t^2 \)
2. \( 154 = u + 25.7 t \)

### Step 3: Solve the System of Equations
Let me solve for the time \(t\) by substituting the second equation into the first.It seems like I can’t do more advanced data analysis right now. Please try again later. 

If you'd like, I can guide you through solving it step-by-step manually, or feel free to reupload the question later!

Here are a few related questions:
1. What is the maximum height the rocket can achieve if it continues accelerating?
2. What would be the time to reach the given height if the acceleration was doubled?
3. How would the final velocity change if the height was 500 meters instead?
4. What happens to the time taken if the initial velocity is zero?
5. How would the time change if gravity played a role in the rocket's motion?

**Tip:** Kinematic equations are very powerful in problems involving constant acceleration; remember to choose the right one based on the available data!

A rocket is launched with some initial velocity and travels with constant acceleration of 25.7 m/s². When the rocket has reached a height of 227 m above its launch position, its velocity is 154 m/s. How long did it take the rocket to reach this height?

Explore how to calculate the time taken for a rocket to reach a height of 227 meters, traveling with constant acceleration of 25.7 m/s² and a final velocity of 154 m/s. This problem uses kinematic equations and system of equations, suitable for high school physics students.

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