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? How long does it take to reach this point?

To solve the problem, we need to find both the velocity of the rocket when it runs out of fuel and the time it takes to reach that altitude.

### Part 1: Velocity of the Rocket When it Runs Out of Fuel

We have already calculated the velocity in the previous step using the kinematic equation:

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

Where:
- \( u = 0 \, \text{m/s} \) (initial velocity)
- \( a = 4.1 \, \text{m/s}^2 \) (acceleration)
- \( s = 860 \, \text{m} \) (altitude)

Substituting the values:

\[
v^2 = 0 + 2 \times 4.1 \times 860 = 7,052 \, \text{m}^2/\text{s}^2
\]

\[
v = \sqrt{7,052} \approx 84 \, \text{m/s}
\]

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

### Part 2: Time to Reach the Altitude of 860 m

To find the time \( t \) it takes to reach the altitude of 860 m, we can use the following kinematic equation:

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

Since the rocket starts from rest (\( u = 0 \)):

\[
860 \, \text{m} = \frac{1}{2} \times 4.1 \, \text{m/s}^2 \times t^2
\]

Simplifying:

\[
860 = 2.05 \, t^2
\]

\[
t^2 = \frac{860}{2.05}
\]

\[
t^2 \approx 419.51
\]

\[
t \approx \sqrt{419.51} \approx 20.48 \, \text{seconds}
\]

So, it takes approximately **20.48 seconds** for the rocket to reach the altitude of 860 m.

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

1. What is the rocket's velocity after 10 seconds of flight?
2. What will be the maximum height the rocket reaches after it runs out of fuel?
3. How long will it take for the rocket to reach its maximum height after running out of fuel?
4. How long will it take for the rocket to fall back to the ground after reaching its maximum height?
5. What would be the rocket's velocity just before it hits the ground?

**Tip:** Always keep track of units in kinematic equations to ensure accuracy in your calculations.

Learn how to calculate the velocity of a rocket when it runs out of fuel and the time it takes to reach an altitude of 860 meters. This problem involves kinematic equations, acceleration due to gravity, and initial conditions of rest. Suitable for high school physics students.

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