Two spacecraft leave Earth on trajectories with the same total energy E < 0 but different angular momenta L₁ > L₂. Which correctly describes their orbits?
ABoth orbits are the same size and shape because energy alone determines the orbit
BBoth are ellipses with the same semi-major axis, but the higher-L orbit has lower eccentricity (more circular)
CThe higher-L orbit is larger because angular momentum determines orbital size
DThe higher-L orbit is hyperbolic because more angular momentum means more kinetic energy
Semi-major axis is determined by energy alone: a = −GMm/(2E). So both orbits have the same semi-major axis and the same period. But angular momentum controls shape (eccentricity): e = √[1 + 2EL²/(μ(GM)²)]. Higher L yields lower e — a more circular ellipse. Lower L yields higher e — a more elongated, needle-like ellipse. This is the key insight: E sets the orbit's size; L sets its shape.
Question 2 Multiple Choice
A comet is observed on a trajectory that will just barely allow it to escape the solar system and reach infinite distance with zero remaining velocity. What type of orbit is it on?
AA hyperbolic orbit, because any escape trajectory must have excess velocity
BA circular orbit, because circular orbits have the minimum energy to remain bound
CA parabolic orbit, because E = 0 corresponds to exactly escaping with zero final velocity
DAn elliptical orbit with very high eccentricity approaching 1
When E = 0, the object has just enough kinetic energy to reach infinity — it escapes the gravitational well but arrives with zero velocity remaining. This is the precise boundary between bound (E < 0, elliptical) and unbound (E > 0, hyperbolic) motion, and it corresponds to a parabolic orbit. A hyperbolic orbit has E > 0 and arrives at infinity with nonzero velocity. An ellipse with e approaching 1 is extremely elongated but still bound (E < 0); it never reaches infinity.
Question 3 True / False
Two objects in the same elliptical orbit but at different positions (one near periapsis, one near apoapsis) have different total energies because their kinetic and potential energies differ at each location.
TTrue
FFalse
Answer: False
Total mechanical energy E = K + U is conserved. As an object moves from apoapsis (slowest, farthest, most negative potential energy contribution) to periapsis (fastest, closest), it gains kinetic energy while its potential energy becomes less negative — the sum stays constant. Both objects have the same total energy E because they are on the same orbit. The semi-major axis a = −GMm/(2E) depends only on E, confirming that all points on the same ellipse share the same total energy.
Question 4 True / False
For a fixed negative total energy, increasing an object's angular momentum makes its orbit more elongated (higher eccentricity).
TTrue
FFalse
Answer: False
The eccentricity formula e = √[1 + 2EL²/(μ(GM)²)] shows that for fixed E < 0 (negative), the term 2EL²/... becomes more negative as L increases, making the expression inside the square root closer to zero — meaning e approaches 0, a more circular orbit. The limiting case L → 0 gives a radial free-fall (e → 1, degenerate ellipse); maximum L gives a circular orbit (e = 0). Higher angular momentum means more 'sideways' motion, which rounds out the orbit.
Question 5 Short Answer
Why do just two conserved quantities — total energy E and angular momentum L — completely determine the shape and size of a gravitational orbit?
Think about your answer, then reveal below.
Model answer: In a two-body gravitational system, energy E = K + U determines whether the orbit is bound (E < 0 → ellipse), marginally bound (E = 0 → parabola), or unbound (E > 0 → hyperbola), and sets the semi-major axis a = −GMm/(2E) for ellipses. Angular momentum L controls how the available energy is distributed between radial and tangential motion, setting the eccentricity e = √[1 + 2EL²/(μ(GM)²)] — how circular versus elongated the orbit is. Together, a and e completely specify the conic section. The gravitational force law (inverse-square) is what makes this exact determination possible; it is a special property of 1/r² forces.
Conservation laws reduce a problem with infinite degrees of freedom (the full trajectory as a function of time) to two numbers that capture the geometry. This is the power of symmetry: the spherical symmetry of gravity conserves angular momentum; the time-invariance of gravity conserves energy. Any orbit consistent with those two conserved values must be a specific conic section — no other trajectory is possible under Newtonian gravity.