Calculate how long a de-orbit burn must last in minutes and seconds to achieve the Shuttle’s change in altitude from 205 miles to 60 miles at perigee.Use the equations and conversions provided below in method 1 or method 2 to find the required burn time
To calculate the required burn time for a de-orbit maneuver, we can use the equations of motion and knowledge of the desired change in altitude. There are two methods to approach this calculation. Let's start with Method 1.
Method 1: Using Equations of Motion
1. Gather the required information:
- Initial altitude = 205 miles
- Final altitude = 60 miles
- Desired change in altitude = Initial altitude - Final altitude
2. Convert the altitudes from miles to feet:
- Initial altitude = 205 miles * 5280 feet/mile
- Final altitude = 60 miles * 5280 feet/mile
3. Calculate the change in altitude in feet:
- Change in altitude = Initial altitude - Final altitude
4. Determine acceleration due to gravity:
- Acceleration due to gravity near the surface of the Earth is approximately 32.2 ft/s^2. However, at orbital altitudes, the gravitational force decreases, so an average value of 30 ft/s^2 is often used.
5. Apply the equation of motion to calculate burn time:
- ΔV (Change in velocity) = Acceleration * Burn time
- ΔV = 2 * Acceleration * Change in altitude
- Burn time = ΔV / (2 * Acceleration)
6. Substitute the values obtained in Step 2, Step 3, and Step 4 into the equation from Step 5, and calculate the burn time.
Method 2: Using Energy Conservation
1. Gather the required information:
- Initial altitude = 205 miles
- Final altitude = 60 miles
2. Convert the altitudes from miles to feet (if not already in feet).
3. Determine the gravitational potential energy (PE) at the initial and final altitudes.
- PE = Mass * Gravitational acceleration * Altitude
4. Apply the energy conservation principle:
- Initial PE - Final PE = Kinetic energy change (KE)
- KE = 0.5 * Mass * Velocity^2
5. Rearrange the equation to solve for velocity:
- Velocity = √((2 * (Initial PE - Final PE)) / Mass)
6. Calculate the velocity using the values obtained in Step 3 and the average mass of the Shuttle.
7. Apply the equation of motion to calculate burn time:
- Burn time = Change in altitude / Velocity
8. Convert the burn time to minutes and seconds if necessary.
By following either Method 1 or Method 2, you should be able to calculate the required burn time to achieve the Shuttle's change in altitude from 205 miles to 60 miles at perigee.