A model rocket is launched straight upward with an initial speed of 48.1 m/s. It accelerates with a constant upward acceleration of 2.15 m/s2 until its engines stop at an altitude of 162 m. How long is the rocket in the air?

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

1. **Powered ascent phase** (while the engines are on and the rocket is accelerating).
2. **Free-fall phase** (after the engines stop and the rocket is only under the influence of gravity).

### **Phase 1: Powered Ascent**

During the powered ascent, the rocket is accelerating upward until it reaches an altitude of 162 m. We can calculate the time it takes to reach this altitude, the final velocity at that altitude, and then move on to the next phase.

#### **Step 1: Time to reach 162 meters**

We can use the following kinematic equation to find the time \( t_1 \) during the powered ascent:

\[
s = v_0 t_1 + \frac{1}{2} a t_1^2
\]

Where:
- \( s = 162 \, \text{m} \) (displacement),
- \( v_0 = 48.1 \, \text{m/s} \) (initial velocity),
- \( a = 2.15 \, \text{m/s}^2 \) (acceleration),
- \( t_1 \) (time during powered ascent).

Plugging in the values:

\[
162 = 48.1 t_1 + \frac{1}{2} \times 2.15 \times t_1^2
\]

This is a quadratic equation of the form \( at^2 + bt + c = 0 \), where:

\[
1.075 t_1^2 + 48.1 t_1 - 162 = 0
\]

#### **Step 2: Solve the quadratic equation**

We can solve this quadratic equation for \( t_1 \) using the quadratic formula:

\[
t_1 = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

Where:
- \( a = 1.075 \),
- \( b = 48.1 \),
- \( c = -162 \).

### **Phase 2: Free-Fall**

Once the engines stop at 162 m, the rocket continues upward under its inertia until gravity brings it to a stop, and then it falls back to the ground.

We need to:
1. **Calculate the maximum altitude** reached.
2. **Find the time to reach maximum height** (this includes the time during free-fall).
3. **Determine the total time of free fall** until the rocket returns to the ground.

Let's proceed with these calculations.The total time the rocket is in the air is approximately **16.76 seconds**.

This includes:
1. The time during the powered ascent phase (while the engines are on).
2. The time to reach the maximum altitude after the engines stop.
3. The time it takes to fall back to the ground.

Would you like more details on any of these steps, or do you have any further questions?

Here are 5 related questions:
1. How high would the rocket go if the engines continued to operate beyond 162 meters?
2. What is the maximum velocity reached by the rocket during the ascent?
3. How would the total time in the air change if the initial velocity were doubled?
4. What would happen to the rocket's trajectory if air resistance were considered?
5. How much kinetic energy does the rocket have when it reaches 162 meters?

**Tip:** When solving kinematic problems, always identify the different phases of motion (e.g., acceleration, constant velocity, free-fall) to apply the correct equations.

A model rocket is launched straight upward with an initial speed of 48.1 m/s. It accelerates with a constant upward acceleration of 2.15 m/s^2 until its engines stop at an altitude of 162 m. How long is the rocket in the air?

Learn how to calculate the total time a model rocket is in the air. This problem involves key kinematic concepts such as constant acceleration, quadratic equations, and free fall. Suitable for high school students, with detailed steps and formulas.

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