Question

Problem B.1: Space Cannon (6 Points)

Scientists are developing a new space cannon to shoot objects from the surface of the Earth directly into a low orbit around the Earth. For testing purposes, a projectile is fired with an initial

velocity of 2.8 km/s vertically into the sky.

Calculate the height that the projectile reaches, ...

(a) assuming a constant gravitational deceleration of 9.81 m/s^2

.

(b) considering the change of the gravitational force with height.

Note: Neglect the air resistance for this problem. Use 6.67×10−11 m^3kg−1

s^−2

for the gravitational

constant, 6371 km for the Earth’s radius, and 5.97 × 1024 kg for the Earth’s mass.

Answer 1

a)

$$v^2=2gh\\(2800)^2=2(9.81)h\\h=400\ km$$

b)

$$0.5v^2=\frac{GM}{R}-\frac{GM}{R+h}\\ 0.5(2800)^2=\frac{(6.67\cdot10^{-11})(5.97\cdot10^{24})}{6.371\cdot10^{6}}\\-\frac{(6.67\cdot10^{-11})(5.97\cdot10^{24})}{6.371\cdot10^{6}+h}\\h=426\ km$$