Training Orbital Mechanics Placement Test Practice — Orbital Mechanics
7 / 7

Placement Test Practice — Orbital Mechanics

25 min Orbital Mechanics

Placement Test Practice — Orbital Mechanics

This practice test covers all major topics in orbital mechanics: Newton's law of gravitation, circular orbits, Kepler's laws, escape velocity, Hohmann transfers, and the vis-viva equation. Each problem requires identifying the correct formula, setting up the calculation, and interpreting the physical significance of the result. These problems integrate concepts from multiple lessons — some require combining gravitational field calculations with orbital mechanics, or energy methods with the vis-viva equation.

Practice Test

1. A new space station is planned at 1000 km altitude. Find its orbital velocity, period, and the number of orbits per day.
2. Calculate the $\Delta v$ for a Hohmann transfer from 200 km altitude to a 12-hour period orbit (semi-synchronous).
3. An elliptical orbit has periapsis at 400 km altitude and apoapsis at 5000 km altitude. Find $a$, $e$, $T$, and the speeds at periapsis and apoapsis.
4. A spacecraft at 500 km altitude has speed 10,500 m/s. Classify the orbit and find $a$.
5. What is the escape velocity from the surface of a neutron star with $M = 2M_{\odot} = 3.978 \times 10^{30}$ kg and $R = 10$ km? Express as a fraction of the speed of light.
6. Use Kepler's third law to find the orbital period of the Hubble Space Telescope (altitude 547 km).
7. A geosynchronous satellite must be moved to a 200 km lower orbit for repair (altitude 35,586 km). Calculate the total $\Delta v$ for the Hohmann transfer.
8. The gravitational acceleration at a planet's surface is 25 m/s² and the escape velocity is 24 km/s. Find the planet's radius.
9. A spy satellite has an elliptical orbit with $a = 7000$ km and $e = 0.05$. What are the minimum and maximum altitudes?
10. Europa orbits Jupiter ($\mu_J = 1.267 \times 10^{17}$ m³/s²) at $r = 671{,}100$ km with $T = 3.551$ days. Verify $\mu_J$ using Kepler's third law.
Show Answer Key

1. $r = 7371$ km. $v = \sqrt{\mu/r} = 7354$ m/s. $T = 2\pi r/v = 6296$ s $= 104.9$ min. Orbits/day $= 1440/104.9 = 13.7$.

2. Semi-sync: $T = 43200$ s. $r_2 = (\mu T^2/4\pi^2)^{1/3} = 26{,}561$ km. $r_1 = 6571$ km. $a_t = 16{,}566$ km. $\Delta v_1 = \sqrt{\mu(2/r_1-1/a_t)} - v_1 = 9521 - 7788 = 1733$ m/s. $\Delta v_2 = v_2 - \sqrt{\mu(2/r_2-1/a_t)} = 3873 - 2357 = 1516$ m/s. Total $= 3249$ m/s.

3. $r_p = 6771$ km, $r_a = 11371$ km. $a = 9071$ km. $e = (11371-6771)/(11371+6771) = 0.254$. $T = 2\pi\sqrt{a^3/\mu} = 8604$ s $= 143.4$ min. $v_p = \sqrt{\mu(2/r_p - 1/a)} = 8708$ m/s. $v_a = \sqrt{\mu(2/r_a - 1/a)} = 5190$ m/s.

4. $r = 6871$ km. $\varepsilon = 10500^2/2 - \mu/r = 5.51 \times 10^7 - 5.80 \times 10^7 = -2.9 \times 10^6$ J/kg. Bound (ellipse). $a = \mu/(2 \times 2.9 \times 10^6) = 6.87 \times 10^7$ m $= 68{,}700$ km.

5. $v_{\text{esc}} = \sqrt{2GM/R} = \sqrt{2 \times 6.674 \times 10^{-11} \times 3.978 \times 10^{30}/10^4} = \sqrt{5.31 \times 10^{16}} = 2.30 \times 10^8$ m/s. That's $v/c = 0.768$ — 77% of the speed of light! (Relativistic effects become significant.)

6. $r = 6371 + 547 = 6918$ km. $T = 2\pi\sqrt{r^3/\mu} = 5735$ s $= 95.6$ min.

7. $r_1 = 42157$ km, $r_2 = 41957$ km. $a_t = 42057$ km. These orbits are so close that $\Delta v$ is tiny: $\Delta v_1 \approx 0.73$ m/s, $\Delta v_2 \approx 0.73$ m/s. Total $\approx 1.5$ m/s.

8. $g = GM/R^2$ and $v_{\text{esc}} = \sqrt{2GM/R}$. $v_{\text{esc}}^2 = 2gR \Rightarrow R = v_{\text{esc}}^2/(2g) = (24000)^2/(2 \times 25) = 11{,}520{,}000$ m $= 11{,}520$ km.

9. $r_p = a(1-e) = 7000(0.95) = 6650$ km. Alt$_p = 6650 - 6371 = 279$ km. $r_a = a(1+e) = 7350$ km. Alt$_a = 979$ km.

10. $\mu = 4\pi^2 r^3/T^2 = 4\pi^2 (6.711 \times 10^8)^3/(3.551 \times 86400)^2 = 1.192 \times 10^{28}/9.414 \times 10^{10} = 1.266 \times 10^{17}$ m³/s². Matches $\mu_J$. ✓

The Vis-Viva Equation and Orbital Energy Back to Module