The shuttle must perform the de-orbit burn to change its orbit so that the perigee, the point in the orbit closest to Earth, is inside of Earth's atmosphere. De-orbit maneuvers are done to lower the perigee of the orbit to 60 miles (or less). An altitude of 60 miles is important because this is where the orbiting spacecraft is recaptured by Earth’s gravity and re-enters Earth’s atmosphere.

Calculate the minimum change in velocity (delta v or ∆v) required for the space shuttle to decrease its altitude to 60 miles when it’s orbiting with an apogee of 246 miles and a perigee of 213 miles above the surface of Earth.

Calculate the minimum change in velocity (delta v or ∆v) required for the space shuttle to decrease its altitude to 60 miles when it’s orbiting with an apogee of 246 miles and a perigee of 213 miles above the surface of Earth.

415

306

perigee changes from 213 to 60

153 miles
multiply 2 due to the rule of 1 mile per 2 ft/s
306 ft/s
it is a slowing maneuver so it is negative
-306 ft/s

Nobody- how did you do that? What formula?

There is no formula. It's simple math. Just look at the units. When you subtract 60 from 213, you get how far you have to travel (still in miles). Then you multiply the result by 2 ft/s per mile. In this, the miles cancel out, and you are left with ft/s. That gives the change in the magnitude of the velocity, but you have to add the direction which is negative because you are decreasing speed rather than increasing it.

To calculate the minimum change in velocity (delta v or ∆v) required for the space shuttle to decrease its altitude to 60 miles, we need to use the principle of conservation of energy. The change in velocity can be determined by comparing the total energy of the shuttle in its current orbit to the total energy required once it reaches the desired altitude.

To start, we can calculate the energy of the shuttle in its current orbit using the specific mechanical energy equation:

E = -(GM) / (2a)

Where:
- E represents the specific mechanical energy
- G is the gravitational constant
- M is the mass of Earth
- a is the semi-major axis of the orbit

Given that the perigee (r1) is 213 miles above the surface of the Earth, and the apogee (r2) is 246 miles above the surface of the Earth, we can calculate the semi-major axis (a) using the following formula:

a = (r1 + r2 + 2rE) / 2

Where:
- rE is the radius of Earth

Let's calculate the semi-major axis (a):

rE = 3960 miles (approximately)

a = (213 + 246 + 2 * 3960) / 2
= 4422 miles

Now we can calculate the specific mechanical energy (E):

E = -(6.67430 x 10^-11 m^3 kg^-1 s^-2) * (5.97219 x 10^24 kg) / (2 * 4422 miles * 1609.34 m/mile)
≈ -3.982 x 10^7 m^2/s^2

Next, let's calculate the energy required for the shuttle to reach an altitude of 60 miles:

rE_new = rE + 60 miles
= 4020 miles (approximately)

a_new = (213 + 60 + 2 * 4020) / 2
= 4153 miles

E_new = -(6.67430 x 10^-11 m^3 kg^-1 s^-2) * (5.97219 x 10^24 kg) / (2 * 4153 miles * 1609.34 m/mile)
≈ -4.199 x 10^7 m^2/s^2

Now, we can calculate the change in energy (ΔE) required:

ΔE = E_new - E
= -4.199 x 10^7 m^2/s^2 - (-3.982 x 10^7 m^2/s^2)
= -2.17 x 10^6 m^2/s^2

The change in velocity (Δv) is equal to the square root of two times the change in energy (ΔE) multiplied by the gravitational constant (G) divided by the mass of the shuttle (m):

Δv = √(2 * ΔE * G / m)

The mass of the space shuttle depends on the specific mission and payload, and will vary. Therefore, we will need to consider it as an unknown variable.

Hence, to calculate the minimum change in velocity (Δv) required for the space shuttle to decrease its altitude to 60 miles, you will need to input the mass of the space shuttle into the equation Δv = √(2 * ΔE * G / m), using the previously obtained value for ΔE.

Note: I have provided the mathematical steps to derive the answer, but since the mass of the space shuttle is not provided, the final calculation cannot be completed without that information.