Question
How do I calculate the maximum altitude reached by a rocket launched at an angle?
Answers
GPT-4o mini
To calculate the maximum altitude reached by a rocket launched at an angle, you will need to consider several factors, including the initial velocity of the rocket, the angle of launch, and the effects of gravity. For simplicity, we will assume that air resistance is negligible.
### Step-by-Step Calculation
1. **Understand the Variables:**
- \( v_0 \): initial launch speed (m/s)
- \( \theta \): launch angle (degrees or radians)
- \( g \): acceleration due to gravity (approximately \( 9.81 \, \text{m/s}^2 \))
2. **Decompose the Initial Velocity:**
The initial velocity can be broken down into its horizontal and vertical components using trigonometric functions:
- Vertical component (\( v_{0y} \)):
\[
v_{0y} = v_0 \cdot \sin(\theta)
\]
- Horizontal component (\( v_{0x} \)): (not needed for maximum altitude)
\[
v_{0x} = v_0 \cdot \cos(\theta)
\]
3. **Calculate the Time to Reach Maximum Height:**
The rocket reaches its maximum height when the vertical component of the velocity becomes zero. We can use the following kinematic equation:
\[
v_{y} = v_{0y} - g \cdot t
\]
Setting \( v_{y} = 0 \) (at the maximum height):
\[
0 = v_{0y} - g \cdot t \implies t = \frac{v_{0y}}{g}
\]
4. **Calculate the Maximum Height:**
Now, we can find the maximum height using the following kinematic equation:
\[
h = v_{0y} \cdot t - \frac{1}{2} g \cdot t^2
\]
Substitute \( t = \frac{v_{0y}}{g} \):
\[
h = v_{0y} \left( \frac{v_{0y}}{g} \right) - \frac{1}{2} g \left( \frac{v_{0y}}{g} \right)^2
\]
Simplifying this expression gives:
\[
h = \frac{(v_{0y})^2}{g} - \frac{1}{2} g \cdot \frac{(v_{0y})^2}{g^2}
\]
\[
h = \frac{(v_{0y})^2}{g} - \frac{(v_{0y})^2}{2g} = \frac{(v_{0y})^2}{2g}
\]
Finally, substitute back for \( v_{0y} \):
\[
h = \frac{(v_0 \cdot \sin(\theta))^2}{2g}
\]
### Final Formula
Thus, the maximum altitude \( h \) reached by the rocket is given by:
\[
h = \frac{(v_0 \cdot \sin(\theta))^2}{2g}
\]
### Key Points
- Make sure to convert degrees to radians if using a calculator that requires radians.
- This calculation assumes no air resistance and that the only force acting on the rocket after launch is gravity. In real-world scenarios, air resistance would reduce the maximum altitude.
### Step-by-Step Calculation
1. **Understand the Variables:**
- \( v_0 \): initial launch speed (m/s)
- \( \theta \): launch angle (degrees or radians)
- \( g \): acceleration due to gravity (approximately \( 9.81 \, \text{m/s}^2 \))
2. **Decompose the Initial Velocity:**
The initial velocity can be broken down into its horizontal and vertical components using trigonometric functions:
- Vertical component (\( v_{0y} \)):
\[
v_{0y} = v_0 \cdot \sin(\theta)
\]
- Horizontal component (\( v_{0x} \)): (not needed for maximum altitude)
\[
v_{0x} = v_0 \cdot \cos(\theta)
\]
3. **Calculate the Time to Reach Maximum Height:**
The rocket reaches its maximum height when the vertical component of the velocity becomes zero. We can use the following kinematic equation:
\[
v_{y} = v_{0y} - g \cdot t
\]
Setting \( v_{y} = 0 \) (at the maximum height):
\[
0 = v_{0y} - g \cdot t \implies t = \frac{v_{0y}}{g}
\]
4. **Calculate the Maximum Height:**
Now, we can find the maximum height using the following kinematic equation:
\[
h = v_{0y} \cdot t - \frac{1}{2} g \cdot t^2
\]
Substitute \( t = \frac{v_{0y}}{g} \):
\[
h = v_{0y} \left( \frac{v_{0y}}{g} \right) - \frac{1}{2} g \left( \frac{v_{0y}}{g} \right)^2
\]
Simplifying this expression gives:
\[
h = \frac{(v_{0y})^2}{g} - \frac{1}{2} g \cdot \frac{(v_{0y})^2}{g^2}
\]
\[
h = \frac{(v_{0y})^2}{g} - \frac{(v_{0y})^2}{2g} = \frac{(v_{0y})^2}{2g}
\]
Finally, substitute back for \( v_{0y} \):
\[
h = \frac{(v_0 \cdot \sin(\theta))^2}{2g}
\]
### Final Formula
Thus, the maximum altitude \( h \) reached by the rocket is given by:
\[
h = \frac{(v_0 \cdot \sin(\theta))^2}{2g}
\]
### Key Points
- Make sure to convert degrees to radians if using a calculator that requires radians.
- This calculation assumes no air resistance and that the only force acting on the rocket after launch is gravity. In real-world scenarios, air resistance would reduce the maximum altitude.