A rocket rises vertically, from rest, with an acceleration of 4.1 m/s2 until it runs out of fuel at an altitude of 860 m . After this point, its acceleration is that of gravity, downward. What is the velocity of the rocket when it runs out of fuel?

To find the velocity of the rocket when it runs out of fuel, we can use the following kinematic equation:

\[
v^2 = u^2 + 2as
\]

Where:
- \( v \) is the final velocity of the rocket.
- \( u = 0 \, \text{m/s} \) is the initial velocity (since the rocket starts from rest).
- \( a = 4.1 \, \text{m/s}^2 \) is the constant acceleration of the rocket.
- \( s = 860 \, \text{m} \) is the altitude (distance) the rocket travels before running out of fuel.

Now, plug in the values:

\[
v^2 = 0^2 + 2 \times 4.1 \, \text{m/s}^2 \times 860 \, \text{m}
\]

\[
v^2 = 2 \times 4.1 \times 860
\]

\[
v^2 = 7,052 \, \text{m}^2/\text{s}^2
\]

\[
v = \sqrt{7,052 \, \text{m}^2/\text{s}^2}
\]

\[
v \approx 84 \, \text{m/s}
\]

So, the velocity of the rocket when it runs out of fuel is approximately **84 m/s**.

Would you like more details or have any questions? Here are five related questions:

1. How long does the rocket take to reach 860 meters?
2. What will be the maximum height the rocket reaches?
3. How much time passes from the rocket running out of fuel to reaching its maximum height?
4. What is the total time of flight for the rocket?
5. How high would the rocket go if its acceleration were doubled?

**Tip:** When solving kinematic problems, always identify which of the five kinematic variables (displacement, initial velocity, final velocity, acceleration, and time) are given and which are required. This helps in selecting the correct equation.

Learn how to calculate the velocity of a rocket when it runs out of fuel using acceleration and distance. This problem involves kinematic equations and is suitable for high school physics students.

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