Question

a. Calculate the velocity that a satellite shot from Newton’s cannon must have to orbit
Earth, 150 km above its surface.
b. How long would it take for the satellite to return to the cannon in seconds and minutes?

Answer 1

a. Calculate the velocity that a satellite shot from Newton's cannon must have to orbit Earth, 150 km above its surface.

b. How long would it take for the satellite to return to the cannon in seconds and minutes?

The second Newton's law for satellite:

$$M a = G \frac {M _ {e} M}{(R + h) ^ {2}}$$ $$a = G \frac {M _ {e}}{(R + h) ^ {2}}$$

Where $a$ is the centripetal acceleration:

$$\frac {v ^ {2}}{R + h} = G \frac {M _ {e}}{(R + h) ^ {2}}$$

So, satellite's speed:

$$v = \sqrt {G \frac {M _ {e}}{R + h}}$$

Using $g = G\frac{M_e}{R^2} \rightarrow GM_e = gR^2$

$$v = R \sqrt {\frac {g}{R + h}}$$ $$v = 6.4 * 10^{6} m \sqrt {\frac {10 m / s ^ {2}}{6 . 4 * 1 0 ^ {6} m + 0 . 1 5 * 1 0 ^ {6} m}} = 7. 9 1 * 1 0 ^ {3} m / s$$

Time to return to the cannon:

$$T = \frac {2 \pi (R + h)}{v}$$ $$T = \frac {2 * 3 . 1 4 * (6 . 4 * 1 0 ^ {6} m + 0 . 1 5 * 1 0 ^ {6} m)}{7 . 9 1 * 1 0 ^ {3} m / s} = 5. 2 0 * 1 0 ^ {3} s$$

Answer: $v = 7.91 * 10^{3}m / s, T = 5.20 * 10^{3}s$