Exercises: Newton's Laws in Space

Work these with a calculator. Use $G = 6.674\times10^{-11}\ \text{N}\cdot\text{m}^2/\text{kg}^2$, Earth's $\mu_\oplus = 3.986\times10^{5}\ \text{km}^3/\text{s}^2 = 3.986\times10^{14}\ \text{m}^3/\text{s}^2$, and $R_\oplus = 6{,}371\ \text{km}$; other bodies' values are in the Section 2.5 table. Difficulty: ⭐ foundational, ⭐⭐ intermediate, ⭐⭐⭐ challenging. Worked solutions to the daggered (†) and odd-numbered problems are in appendices/answers-to-selected.md — try each cold before you peek. For the "implement it" problems, do not run the code: hand-trace it and write the result in an # Expected output: comment, exactly as the chapter does. Keep units on every quantity and sanity-check every number.

Part A — Warm-ups: laws, momentum, thrust (⭐)

2.1 † A $12{,}000\ \text{kg}$ upper stage coasts at $7.8\ \text{km/s}$. What is its momentum, in $\text{kg}\cdot\text{m/s}$?

2.2 An astronaut of total suited mass $100\ \text{kg}$, at rest relative to their spacecraft, throws a $5\ \text{kg}$ tool bag at $4\ \text{m/s}$. What is the astronaut's recoil speed, and in which direction?

2.3 † A rocket engine expels propellant at $\dot m_p = 300\ \text{kg/s}$ with an effective exhaust velocity of $v_e = 3{,}100\ \text{m/s}$. What thrust does it produce? Give the answer in newtons and in kilonewtons.

2.4 An engine must produce $800\ \text{kN}$ of thrust at an exhaust velocity of $3{,}200\ \text{m/s}$. What propellant mass flow rate $\dot m_p$ does that require?

2.5 † Using $\mu_{\text{Mars}} = 4.283\times10^{4}\ \text{km}^3/\text{s}^2$ and $R_{\text{Mars}} = 3{,}390\ \text{km}$, compute the escape velocity from the surface of Mars.

2.6 Compute the surface gravity of the Moon from $\mu_{\text{Moon}} = 4.903\times10^{3}\ \text{km}^3/\text{s}^2$ and $R_{\text{Moon}} = 1{,}737\ \text{km}$. (Watch your units: convert to metres, or keep everything in km and convert the final answer.) Does your value match the "one-sixth $g$" folklore?

Part B — Gravitation and the gravitational parameter (⭐⭐)

2.7 † A geostationary satellite orbits at a radius of $r = 42{,}164\ \text{km}$ from Earth's centre. How strong is Earth's gravity there, in $\text{m/s}^2$, and what fraction of surface gravity is that?

2.8 Mars's mass is about $6.417\times10^{23}\ \text{kg}$. Compute its gravitational parameter $\mu = GM$ and confirm it matches the Section 2.5 table. Why do mission planners quote $\mu$ rather than $M$?

2.9 † At what altitude above Earth's surface does gravitational acceleration fall to exactly half its surface value? (Set $\mu/r^2 = \tfrac12\,\mu/R_\oplus^2$ and solve for $r$, then subtract $R_\oplus$.)

Part C — Escape velocity and energy (⭐⭐)

2.10 † Compute the escape velocity from the Moon's surface, and express it as a fraction of Earth's $11.2\ \text{km/s}$. Explain in one sentence why a low escape velocity made the Apollo lunar ascent stage so much simpler than an Earth launcher.

2.11 Starting from $v_{\text{esc}} = \sqrt{2\mu/r}$ and the circular orbital speed $v_{\text{circ}} = \sqrt{\mu/r}$, show algebraically that $v_{\text{esc}} = \sqrt{2}\,v_{\text{circ}}$ at any radius. By what percentage does escape exceed orbit?

2.12 † The depth of a body's gravity well, in energy per kilogram, is $\mu/R$. Compute it for Mars and show that it equals $\tfrac12\,v_{\text{esc}}^2$ using your answer from 2.5.

Part D — Implement it in Python (⭐⭐)

Write each function, then hand-trace it for the given inputs and record the result in an # Expected output: comment. Do not run it.

2.13 † Write escape_velocity(mu, r) returning $\sqrt{2\mu/r}$ (SI units). Trace it for escape_velocity(3.986e14, 6.371e6).

2.14 Write surface_gravity(mu, r) returning $\mu/r^2$. Trace it for surface_gravity(3.986e14, 6.371e6) and check it against $9.81\ \text{m/s}^2$.

2.15 † Write separation_velocity(m_total, m_ejected, ve) that returns the forward speed a spacecraft gains when it ejects m_ejected of gas at exhaust velocity ve (relative to the craft) in a single puff, using momentum conservation: $V = m_{\text{ejected}}\,v_e / m_{\text{total}}$. Trace it for separation_velocity(1000, 10, 3000).

Part E — Find the error (⭐⭐)

2.16 † A student writes Newton's second law for a rocket as $F = \dfrac{d(mv)}{dt} = m\dot v + v\,\dot m$ and declares "the thrust is $v\,\dot m$, so a rocket standing still on the pad ($v = 0$) produces no thrust." Identify the two physical errors and give the correct expression for thrust.

2.17 A popular article states: "Astronauts on the ISS are weightless because, at $420\ \text{km}$ up, they have escaped Earth's gravity." Compute the actual gravity at that altitude and explain, in terms of free fall, why the article's reasoning is wrong even though the astronauts really do float.

Part F — Design it and back of the envelope (⭐⭐ / ⭐⭐⭐)

2.18 † (Mission project) For your chosen mission (Track A/B/C/D), identify the primary body your spacecraft orbits or lands on, and record its $\mu$, mean radius, surface gravity, and surface escape velocity in your MDR's "Primary Body" data block. State one design consequence of that escape velocity (e.g., how hard it is to leave that body).

2.19 Back of the envelope. Low Earth orbital speed is about $7.7\ \text{km/s}$. Without a detailed calculation, estimate the escape velocity from that same altitude, and compare it with the $11.2\ \text{km/s}$ from the surface. Why is escaping from orbit cheaper than escaping from the ground?

2.20 † Back of the envelope. The ISS has a mass of about $420{,}000\ \text{kg}$ and orbits at roughly $7.66\ \text{km/s}$. Estimate its orbital momentum. Then estimate the impulse (in $\text{N}\cdot\text{s}$) a reboost must supply to raise its speed by $1\ \text{m/s}$, and comment on how that compares with a $3\ \text{kN}$ thruster firing for a few minutes.

2.21 Design it. You need a thruster that produces $500\ \text{kN}$ of thrust using a propellant with $v_e = 3{,}000\ \text{m/s}$. What mass flow rate is required, and how much propellant is consumed in a $120\ \text{s}$ burn? (Use $T = \dot m_p v_e$.)

Part G — Why can't you just… and interleaved (⭐⭐⭐)

2.22 † Why can't you just put a big propeller or fan on a spacecraft to fly through space? Answer in terms of Newton's third law and what a propeller actually pushes against.

2.23 Why can't you just climb high enough to "get above" Earth's gravity and float away? Use the inverse-square law to show that gravity never reaches zero, and estimate how far out (in Earth radii) you must go for it to drop to $1\%$ of its surface value.

2.24 † (Ch. 1, interleaved) Chapter 1 says reaching LEO costs about $9.4\ \text{km/s}$ of delta-v, but the orbital speed there is only about $7.8\ \text{km/s}$. Name the two effects that account for the extra $\sim 1.6\ \text{km/s}$, and say which of Newton's laws each is fundamentally about. (You will quantify them in Chapter 4.)

2.25 (Ch. 1, interleaved) Chapter 1 called orbit a "microgravity" environment. Given that gravity at the ISS is about $88\%$ of its surface value, explain in two or three sentences why astronauts nonetheless float, and why "microgravity" is a more honest word than "zero gravity."

2.26 † (Synthesis) In one short paragraph, explain how the conservation of momentum you proved in §2.4 becomes the rocket equation of Chapter 3. What has to happen "a little at a time, continuously" for the one-line momentum result to turn into $\Delta v = v_e \ln(m_0/m_f)$?

Solutions to † and odd-numbered exercises are in appendices/answers-to-selected.md. Reference code for the "implement it" problems is in code/exercise-solutions.py. For design and back-of-the-envelope problems, the rubric rewards: a clearly stated assumption, explicit units throughout, and a final sanity check on the number.