Self-Assessment Quiz: Energy in Space — The Vis-Viva Equation

Twenty questions to check your grasp of orbital energy, the vis-viva equation, escape, and the "higher-is-slower" paradox. Answer each before opening the key. Aim for 16 or more. Use $\mu_\oplus = 3.986\times10^{5}\ \text{km}^3/\text{s}^2$ and $R_\oplus = 6{,}371\ \text{km}$.

Question 1

The specific orbital energy $\varepsilon = \tfrac{v^2}{2} - \tfrac{\mu}{r}$ of a coasting spacecraft is:

A) always positive B) constant (conserved) as it coasts around its orbit C) larger for a heavier spacecraft in the same orbit D) equal to its kinetic energy

Question 2

The vis-viva equation $v^2 = \mu\left(\tfrac{2}{r} - \tfrac{1}{a}\right)$ gives you:

A) the orbital period B) the speed at any point on any orbit C) the escape velocity and nothing else D) the mass ratio of the rocket

Question 3

For any bound orbit (a closed ellipse or circle), the specific orbital energy is:

A) positive B) zero C) negative D) infinite

Question 4

Compared with a lower circular orbit, a higher circular orbit has:

A) higher speed B) lower speed C) exactly the same speed D) zero speed

Question 5

The circular orbital velocity at radius $r$ is:

A) $\sqrt{2\mu/r}$ B) $\sqrt{\mu/r}$ C) $\mu/r$ D) $\sqrt{\mu/(2r)}$

Question 6

Escape velocity at a given radius equals the circular velocity there multiplied by:

A) $2$ B) $\sqrt{2}$ C) $\tfrac{1}{2}$ D) $\pi$

Question 7

A parabolic (marginal-escape) trajectory has a specific orbital energy of:

A) less than zero B) exactly zero C) greater than zero D) undefined

Question 8

A hyperbolic trajectory has a semi-major axis $a$ that is:

A) positive B) zero C) negative D) always equal to $r$

Question 9

In $\varepsilon = -\mu/(2a)$, an orbit with a larger semi-major axis has:

A) lower (more negative) energy B) higher (less negative) energy C) exactly zero energy D) positive energy in every case

Question 10

To catch up with a target that is ahead of you in the same circular orbit, you should burn:

A) prograde (toward the target) B) retrograde (away from the target) C) straight up (radially outward) D) not at all — you will drift into it

Question 11

The orbital speed in a $400\ \text{km}$ circular LEO is about:

A) $3.1\ \text{km/s}$ B) $7.7\ \text{km/s}$ C) $11.2\ \text{km/s}$ D) $27\ \text{km/s}$

Question 12

The orbital speed in geostationary orbit (GEO) is about:

A) $3.1\ \text{km/s}$ B) $7.7\ \text{km/s}$ C) $11.2\ \text{km/s}$ D) $1.0\ \text{km/s}$

Question 13

On an elliptical orbit, the spacecraft moves fastest at:

A) apoapsis (the highest point) B) periapsis (the lowest point) C) the midpoint of the ellipse D) nowhere — the speed is constant

Question 14

In the vis-viva equation, the symbol $r$ stands for:

A) the semi-major axis of the orbit B) the spacecraft's current distance from the center of the body C) the periapsis distance, always D) the radius of the planet

Question 15 (True/False, justify)

"Raising a satellite to a higher orbit increases both its energy and its speed." True or false? Justify in one sentence.

Question 16 (True/False, justify)

"A spacecraft's specific orbital energy $\varepsilon$ depends on the spacecraft's mass." True or false? Explain briefly.

Question 17 (True/False, justify)

"Escape velocity from a $400\ \text{km}$ low Earth orbit is about $11.2\ \text{km/s}$." True or false? Say what is right and what is wrong about the statement.

Question 18 (Short answer)

A satellite is in a circular orbit at $r = 6{,}771\ \text{km}$. Compute (a) its circular speed and (b) its specific orbital energy $\varepsilon$ in $\text{MJ/kg}$. Show the two calculations.

Question 19 (Short answer)

Explain, in two sentences, why firing prograde (speeding up your instantaneous velocity) leaves you in an orbit where you ultimately move slower. Refer to where the added energy goes.

Question 20 (Short answer)

State this chapter's threshold concept in your own words, and give one practical consequence of it for flying a rendezvous.

Answer Key

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