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 246 miles and a perigee of 212 miles above the surface of Earth

To calculate the minimum change in velocity (delta V or ∆V) required for the Space Shuttle to decrease its altitude to 60 miles, you can use the concept of orbital mechanics and the vis-viva equation.

The vis-viva equation relates the orbit's semi-major axis (a), the distance from the center of the object being orbited (r), and the velocity (v) of the orbiting object. It is derived from the conservation of mechanical energy for an object in a gravitational field.

The vis-viva equation is given as:

v^2 = GM * (2/r - 1/a)

Where:
- v is the velocity of the orbiting object
- G is the gravitational constant
- M is the mass of the object being orbited
- r is the distance from the center of the object being orbited
- a is the semi-major axis of the orbit

In this case, we are given the apogee (maximum distance from Earth) and perigee (minimum distance from Earth) of the orbit, which allows us to find the semi-major axis (a). The semi-major axis (a) can be calculated as the average of the apogee and perigee:

a = (apogee + perigee) / 2

Given:
- Apogee = 246 miles
- Perigee = 212 miles

a = (246 + 212) / 2
a = 458 / 2
a = 229 miles

Now we can calculate the velocity of the orbiting object at its current altitude using the vis-viva equation:

v = sqrt(GM * (2/r - 1/a))

Next, we need to calculate the velocity required for the new altitude of 60 miles. We will use the same equation, substituting the new altitude into the equation:

v_new = sqrt(GM * (2/r_new - 1/a))

Given:
- r_new = 60 miles

Now we need to calculate the change in velocity (∆V) required. The change in velocity (∆V) is given by:

∆V = |v_new - v|

Thus, to find the minimum change in velocity (delta V or ∆V) required for the Space Shuttle to decrease its altitude to 60 miles, you will need to plug in the values into the equations above.