A model rocket is launched straight upward with an initial speed of 49.8 ,/s. it accelerates wtih a constant upward acceleration of 1.78 m/s^2 until its engines stop at an altitude of 195.1 m. how long is the rocket in the air.

To determine how long the rocket is in the air, we can break the problem into two phases: 

1. **Phase 1 (Powered flight):** The rocket accelerates upward until its engines stop at a height of 195.1 m.
2. **Phase 2 (Free fall):** After the engines stop, the rocket moves upwards until it reaches its peak, then falls back to the ground under the influence of gravity.

### Phase 1: Powered flight
We can use the kinematic equation to find the time during powered flight:
\[
h = v_0 t + \frac{1}{2} a t^2
\]
where:
- \( h = 195.1 \, \text{m} \) is the altitude at which the engines stop,
- \( v_0 = 49.8 \, \text{m/s} \) is the initial speed,
- \( a = 1.78 \, \text{m/s}^2 \) is the constant upward acceleration,
- \( t \) is the time during powered flight.

Let's solve for \( t \) using this equation.

### Phase 2: Free fall
In this phase, the rocket moves upwards with an initial velocity equal to the velocity at the end of Phase 1, and then falls back down under the effect of gravity. The time to reach the peak can be found by considering the upward velocity at the start of free fall. Then we calculate the time it takes to fall back to the ground. 

Let's solve for these values.The time during the powered flight is approximately \( 3.68 \, \text{seconds} \). We discard the negative solution as time cannot be negative.

Next, we need to calculate the velocity at the moment when the engines stop, then use this to calculate the rest of the time the rocket is in the air. Let me proceed with these calculations.It seems like I can’t do more advanced data analysis right now. Please try again later. If you'd like, I can walk you through the remaining steps of the problem or help with any other questions.

Would you like to continue, or should I help with something else?

Here are five related questions you might find useful:
1. How do we calculate the maximum height a projectile reaches?
2. What are the key kinematic equations for uniformly accelerated motion?
3. How does gravity affect the total time a projectile stays in the air?
4. How would the initial speed affect the total flight time of the rocket?
5. What happens to the rocket's velocity right before it hits the ground?

**Tip:** In projectile motion problems, always split the problem into distinct phases, such as upward and downward motion, for easier analysis.

A model rocket is launched straight upward with an initial speed of 49.8 m/s. It accelerates with a constant upward acceleration of 1.78 m/s^2 until its engines stop at an altitude of 195.1 m. How long is the rocket in the air?

Find out how to calculate the total time a rocket stays in the air after being launched with an initial speed of 49.8 m/s and an acceleration of 1.78 m/s². The rocket's engines stop at an altitude of 195.1 m. Learn how to apply kinematic equations to solve this physics problem, suitable for Grades 10-12.

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