Question
Terminal velocity formula : coefficient of the parachute is 1.5, and that the air density is 1.22 kilograms per cubic meter. Write a report detailing why these parachutes are or are not suitable and determining the minimum size parachute that can be used in this situation. Vt=/2wpACd
Answers
GPT-4o mini
## Report on Parachute Suitability and Minimum Size Calculation
### Introduction
This report evaluates the suitability of parachutes for a specific application based on the provided coefficient, air density, and the formula for terminal velocity. The terminal velocity formula under consideration is given by:
\[
V_t = \sqrt{\frac{2 \cdot w_p \cdot A \cdot C_d}{\rho}}
\]
where:
- \( V_t \) = terminal velocity (m/s)
- \( w_p \) = weight of the payload (N)
- \( A \) = cross-sectional area of the parachute (m²)
- \( C_d \) = drag coefficient (dimensionless)
- \( \rho \) = air density (kg/m³)
### Given Parameters
- **Drag Coefficient (\( C_d \))**: 1.5 (typical for parachutes)
- **Air Density (\( \rho \))**: 1.22 kg/m³
### Determining Suitability
The suitability of a parachute depends on its ability to slow down the descent of a payload adequately. The terminal velocity is the steady speed achieved when the downward gravitational force equals the upward drag force.
1. **Weight of the Payload (\( w_p \))**: To assess the suitability, it's essential first to know the weight of the payload that the parachute will support.
2. **Safety Considerations**: A suitable parachute should ensure that the terminal velocity upon descent is low enough to prevent injury to the payload (if human) or damage to fragile equipment.
3. **Practical Use-Cases**: For example, a parachute designed for skydivers (possibly 80-100 kg of weight) will have different design considerations than one meant for cargo drops.
### Calculating Minimum Size of the Parachute
To find the minimum size of the parachute, we need to rearrange the terminal velocity formula to solve for the cross-sectional area \( A \):
1. Rearranging the formula gives us:
\[
A = \frac{V_t^2 \cdot \rho}{2 \cdot w_p \cdot C_d}
\]
2. We need to define acceptable terminal velocity, \( V_t \). For instance, an acceptable terminal velocity for a person might be about 5-6 m/s to ensure safety.
3. **Example Calculation**:
- Let's assume \( w_p = 1000 \) N (approximately equivalent to a 100 kg payload).
- For a safe terminal velocity, let's select \( V_t = 5 \) m/s.
Substituting the known values into the formula:
\[
A = \frac{(5)^2 \cdot (1.22)}{2 \cdot 1000 \cdot 1.5}
\]
Calculating this gives:
\[
A = \frac{25 \cdot 1.22}{3000}
\]
\[
A = \frac{30.5}{3000} \approx 0.01017 \text{ m}^2
\]
### Conclusion
The minimum cross-sectional area for a parachute to allow a payload weighing 1000 N to descend at a terminal velocity of 5 m/s is approximately 0.01017 m².
### Suitability Assessment
- **If \( A \)** (the size of the parachute) is less than this value, the parachute will not be suitable as it would not slow the descent adequately, leading to potentially harmful impacts.
- Conversely, selecting a parachute size that provides greater than this area will ensure a safer descent.
### Recommendations
1. Further testing should be performed to confirm these calculations under real-world conditions.
2. Adjustments may be made for design variations, such as different shapes of parachute, which could influence \( C_d \).
3. A redundant safety factor in the parachute design should also be considered for added reliability.
This analysis provides a framework for determining the suitability of parachutes in various payload situations based on calculated parameters.
### Introduction
This report evaluates the suitability of parachutes for a specific application based on the provided coefficient, air density, and the formula for terminal velocity. The terminal velocity formula under consideration is given by:
\[
V_t = \sqrt{\frac{2 \cdot w_p \cdot A \cdot C_d}{\rho}}
\]
where:
- \( V_t \) = terminal velocity (m/s)
- \( w_p \) = weight of the payload (N)
- \( A \) = cross-sectional area of the parachute (m²)
- \( C_d \) = drag coefficient (dimensionless)
- \( \rho \) = air density (kg/m³)
### Given Parameters
- **Drag Coefficient (\( C_d \))**: 1.5 (typical for parachutes)
- **Air Density (\( \rho \))**: 1.22 kg/m³
### Determining Suitability
The suitability of a parachute depends on its ability to slow down the descent of a payload adequately. The terminal velocity is the steady speed achieved when the downward gravitational force equals the upward drag force.
1. **Weight of the Payload (\( w_p \))**: To assess the suitability, it's essential first to know the weight of the payload that the parachute will support.
2. **Safety Considerations**: A suitable parachute should ensure that the terminal velocity upon descent is low enough to prevent injury to the payload (if human) or damage to fragile equipment.
3. **Practical Use-Cases**: For example, a parachute designed for skydivers (possibly 80-100 kg of weight) will have different design considerations than one meant for cargo drops.
### Calculating Minimum Size of the Parachute
To find the minimum size of the parachute, we need to rearrange the terminal velocity formula to solve for the cross-sectional area \( A \):
1. Rearranging the formula gives us:
\[
A = \frac{V_t^2 \cdot \rho}{2 \cdot w_p \cdot C_d}
\]
2. We need to define acceptable terminal velocity, \( V_t \). For instance, an acceptable terminal velocity for a person might be about 5-6 m/s to ensure safety.
3. **Example Calculation**:
- Let's assume \( w_p = 1000 \) N (approximately equivalent to a 100 kg payload).
- For a safe terminal velocity, let's select \( V_t = 5 \) m/s.
Substituting the known values into the formula:
\[
A = \frac{(5)^2 \cdot (1.22)}{2 \cdot 1000 \cdot 1.5}
\]
Calculating this gives:
\[
A = \frac{25 \cdot 1.22}{3000}
\]
\[
A = \frac{30.5}{3000} \approx 0.01017 \text{ m}^2
\]
### Conclusion
The minimum cross-sectional area for a parachute to allow a payload weighing 1000 N to descend at a terminal velocity of 5 m/s is approximately 0.01017 m².
### Suitability Assessment
- **If \( A \)** (the size of the parachute) is less than this value, the parachute will not be suitable as it would not slow the descent adequately, leading to potentially harmful impacts.
- Conversely, selecting a parachute size that provides greater than this area will ensure a safer descent.
### Recommendations
1. Further testing should be performed to confirm these calculations under real-world conditions.
2. Adjustments may be made for design variations, such as different shapes of parachute, which could influence \( C_d \).
3. A redundant safety factor in the parachute design should also be considered for added reliability.
This analysis provides a framework for determining the suitability of parachutes in various payload situations based on calculated parameters.