Exercises: Getting to Orbit

Work these with a calculator, $g_0 = 9.81\ \text{m/s}^2$, and Earth's $\mu = 3.986\times10^5\ \text{km}^3/\text{s}^2$, $R_\oplus = 6{,}371\ \text{km}$. Difficulty: ⭐ foundational, ⭐⭐ intermediate, ⭐⭐⭐ challenging. Worked solutions to the daggered (†) and odd-numbered problems are in the appendix answers-to-selected.md — try each one 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.

Part A — Warm-ups: orbit is sideways (⭐)

4.1 † Compute the circular orbital speed at an altitude of $300\ \text{km}$ (so $r = 6{,}671\ \text{km}$) using $v = \sqrt{\mu/r}$. Is it larger or smaller than the $7.9\ \text{km/s}$ at the surface?

4.2 A launch vehicle must supply an orbital speed of $7.8\ \text{km/s}$ plus $1.5\ \text{km/s}$ of gravity loss and $0.1\ \text{km/s}$ of drag loss. What is its total launch delta-v (no rotation credit)?

4.3 † In one sentence each, explain why the launch delta-v (~$9.4\ \text{km/s}$) is larger than the orbital speed (~$7.8\ \text{km/s}$), and name the two loss terms responsible.

4.4 A suborbital vehicle just reaches the Kármán line at $100\ \text{km}$. Using $v = \sqrt{2gh}$ (constant $g$), estimate the ideal vertical launch speed it needs.

4.5 † A body in low orbit moves about $7.9\ \text{km}$ sideways each second. Using $\text{drop} = d^2/(2R)$, find how far Earth's surface curves away beneath it in that second, and compare with how far a free body falls in one second. What does the comparison mean?

4.6 State, in one sentence each, the definitions of a gravity turn and a pitch program.

Part B — Gravity and drag losses (⭐⭐)

4.7 † A first stage burns for $160\ \text{s}$. If it flew perfectly vertically the entire time, what gravity loss would it accumulate? Why is the loss of a real gravity turn smaller?

4.8 The instantaneous gravity-loss rate is $g\sin\gamma$, where $\gamma$ is the flight-path angle above horizontal. Compute the loss rate at $\gamma = 90^\circ$, $\gamma = 30^\circ$, and $\gamma = 0^\circ$. Which costs the most, and which is "free"?

4.9 † During a landing test a rocket hovers (thrust-to-weight $= 1$) for $5\ \text{s}$. How much delta-v does it spend to gain zero speed and zero altitude? Explain how this "hover trap" is why lift-off thrust-to-weight must exceed 1.

4.10 At max-Q the dynamic pressure is $q = \tfrac12\rho v^2 \approx 30\ \text{kPa}$. For a vehicle of frontal area $A = 10.75\ \text{m}^2$, drag coefficient $C_d = 0.4$, and mass $360\ \text{t}$, find the drag force $D = qC_dA$ and the drag deceleration $D/m$. Compare it with $g$.

4.11 † Drag grows as $\rho v^2$, and a rocket reaches Mach 20+. Explain in physical terms why the total drag loss (~$0.1\ \text{km/s}$) is nonetheless far smaller than the gravity loss (~$1.5\ \text{km/s}$). (Hint: when is $\rho$ large, and when is $v$ large?)

Part C — 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. Reference solutions are in code/exercise-solutions.py.

4.12 † Write circular_velocity(mu, r) returning $\sqrt{\mu/r}$. Trace it for circular_velocity(3.986e5, 6771) (units: km, km/s).

4.13 Write launch_delta_v(v_orbit, grav, drag, rot=0.0) returning $v_{\text{orbit}} + \text{grav} + \text{drag} - \text{rot}$. Trace it for launch_delta_v(7800, 1500, 100, 0) and for launch_delta_v(7800, 1500, 100, 465).

4.14 † Write gravity_loss(g, sin_gamma_avg, t_burn) returning $g\cdot\langle\sin\gamma\rangle\cdot t_b$. Trace it for gravity_loss(9.81, 0.6, 162) and for the straight-up bound gravity_loss(9.81, 1.0, 162).

Part D — Find the error (⭐⭐)

4.15 † A student writes: "To reach the ISS at $400\ \text{km}$, I only need enough delta-v to climb $400\ \text{km}$: $v = \sqrt{2gh} = \sqrt{2\times9.81\times400{,}000} \approx 2.8\ \text{km/s}$. The rest of the $9.4\ \text{km/s}$ is wasted." Diagnose the error in physical terms.

4.16 A blog claims: "Flying straight up is the most efficient path to space, because a straight line is the shortest path." Identify the flaw, using the gravity-loss integral.

Part E — Design it (⭐⭐ / ⭐⭐⭐)

4.17 † Build a launch delta-v budget for a due-east launch to a $7.8\ \text{km/s}$ LEO from a site on the equator, including the Earth-rotation credit of $0.46\ \text{km/s}$. Compare the total with the no-credit figure and comment on why launch sites favor low latitudes.

4.18 (Mission project) Add a "launch to LEO ≈ $9.4\ \text{km/s}$" line to your mission's delta-v budget (Track A/B/C/D). Note whether the launch vehicle or your spacecraft pays it, and record it in your MDR.

4.19 † Vehicle X lifts off at thrust-to-weight $1.2$; vehicle Y at $1.5$. Argue qualitatively which suffers more gravity loss and which suffers more drag/max-Q stress, and why a designer must choose a value between the extremes.

Part F — Back of the envelope & "why can't you just…" (⭐⭐⭐)

4.20 † Estimate the ratio of the energy needed to reach orbit ($v \approx 7.8\ \text{km/s}$) to the energy needed to reach $100\ \text{km}$ suborbital ($v \approx 1.4\ \text{km/s}$). State clearly what your comparison implies about "reaching space" versus "reaching orbit."

4.21 Why can't you just fly a high-altitude balloon or a jet up to the edge of space and drift into orbit? Frame your answer entirely in terms of the missing $\sim 7.8\ \text{km/s}$ of sideways speed.

4.22 † Why can't you just rise very slowly and gently, hovering upward, to save fuel? Answer using the gravity-loss integral and the hover trap.

Part G — Interleaved & synthesis (from Chapters 1–3)

4.23 † (Ch. 3) A launch vehicle must supply $9.4\ \text{km/s}$ with an effective $v_e = 3.0\ \text{km/s}$. Find the mass ratio it needs, compare with the single-stage ceiling $1/\varepsilon \approx 12.5$ from Chapter 3, and state what this forces.

4.24 (Ch. 2) Earth's surface escape velocity is $\sim 11.2\ \text{km/s}$ and circular orbital velocity is $\sim 7.9\ \text{km/s}$. Show that the ratio is $\sqrt{2}$, starting from $v_{\text{esc}} = \sqrt{2\mu/r}$ and $v_{\text{circ}} = \sqrt{\mu/r}$.

4.25 † (Ch. 1, synthesis) Chapter 1 quoted $\sim 9.4\ \text{km/s}$ to reach orbit; Chapter 3 quoted orbital speed as $\sim 7.8\ \text{km/s}$. Reconcile the two numbers precisely, using this chapter's loss accounting.

4.26 (Ch. 2 / Ch. 3 synthesis) In Newton's cannonball, at what horizontal speed must the ball leave a frictionless, airless mountaintop at the surface to enter a circular orbit? Which chapter's physics gives that speed, and does the rocket equation care how the ball reached it?

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 problems, the rubric rewards: correct set-up of the orbital speed and loss terms, explicit units, an explicit statement of whether a delta-v is ideal or includes losses, and a sanity check on the final number.