A rocket is fired straight up through the atmosphere from the South Pole, burning out at an altitude of 208 km when traveling at 5.8 km/s.

(a) What maximum distance from Earth's surface does it travel before falling back to Earth?

To find the maximum distance from Earth's surface that the rocket travels before falling back, we need to calculate the highest point it reaches from the surface.

We know that the rocket burns out at an altitude of 208 km, which means it reaches a maximum height of 208 km above the Earth's surface.

To find the maximum distance from the Earth's surface, we need to add the radius of the Earth to the maximum height. The radius of the Earth is approximately 6,371 km.

Therefore, the maximum distance from the Earth's surface that the rocket travels before falling back is:

208 km + 6,371 km = 6,579 km

To find the maximum distance from Earth's surface that the rocket travels before falling back, we need to determine the height at its highest point and then subtract the radius of the Earth.

The total height reached by the rocket is the sum of the initial height from the South Pole and the altitude at which the rocket burns out. The initial height is zero (since the rocket starts from the Earth's surface).

Therefore, the maximum distance from Earth's surface is:
Maximum Distance = Total Height - Earth's radius

1. Calculate the total height:
Total Height = Initial Height + Altitude at burnout

Since the initial height is 0,
Total Height = 0 + 208 km = 208 km

2. Determine Earth's radius:
The approximate average radius of the Earth is 6,371 km.

3. Calculate the maximum distance from Earth's surface:
Maximum Distance = Total Height - Earth's radius
Maximum Distance = 208 km - 6,371 km

So, the maximum distance from Earth's surface that the rocket travels before falling back is approximately 6,163 km.

The Earth's radius is Re = 6378 km.

Burnout occurs when R = 6586 km.

At the highest altitude, the burnout kinetic energy equals the gain in potential energy.
(1/2) M V^2 = GM[1/R - 1/(R+h)]

Solve for R+h. R+h-Re is the distance from the surface of the earth.

M is the Earth's mass and G is the universal constant of gravity.

In case you don't want to look up M and G,you can use the fact that

GM/Re^2 = g = 9.81 m/s^2