Calculate the minimum change in velocity (delta V or ∆V) required for the Space Shuttle to decrease its altitude to 60 miles if it’s orbiting with an apogee of 260 miles and a perigee of 220 miles above the surface of Earth.
Do the VASTS module yourself!
To calculate the minimum change in velocity required for the Space Shuttle to decrease its altitude, we need to consider the principles of orbital mechanics. Here's how you can approach this problem:
1. Determine the initial velocity (V1) of the Space Shuttle in its current orbit. We can use the formula for velocity in circular orbit: V = √(GM/R), where G is the gravitational constant, M is the mass of the Earth, and R is the distance from the center of the Earth to the Shuttle.
- G = 6.67430 x 10^-11 m^3/(kg s^2) (Gravitational Constant)
- M = 5.972 × 10^24 kg (Mass of the Earth)
- R = average of the apogee and perigee distances from the Earth's center (altitude + radius of Earth)
2. Calculate the final velocity (V2) required for the Shuttle to achieve the desired altitude of 60 miles above the Earth's surface. Similar to the step above, use the same formula to calculate the velocity.
3. Calculate the change in velocity (∆V) by subtracting the initial velocity (V1) from the final velocity (V2). This will give us the minimum change in velocity required.
Let's perform the calculations:
Step 1: Calculating Initial Velocity (V1)
- Radius of Earth = 3959 miles
- R = average of apogee (260 miles + 3959 miles) and perigee (220 miles + 3959 miles)
- Calculate V1 using the formula V = √(GM/R)
Step 2: Calculating Final Velocity (V2)
- R = 3959 miles + 60 miles (desired altitude)
- Calculate V2 using the formula V = √(GM/R)
Step 3: Calculate the change in velocity (∆V) by subtracting V1 from V2.
Performing these calculations will give us the minimum change in velocity (∆V) required for the Space Shuttle to decrease its altitude to 60 miles.