Worked Solutions
1. Estimating the Tunguska Event Energy
Mass = (4/3)π(30 m)³(2000 kg/m³) = 2.262e+08 kg. Energy = ½mv² = ½(2.262e+08)(15000)² = 2.545e+16 J = 6.1 MT. This result (6.1 MT) is broadly consistent with, though somewhat below, the commonly cited 3-15 MT range, which reflects genuine uncertainty in the actual impactor size and airburst altitude, not an error in the basic kinetic-energy calculation.
| Criterion | Points |
|---|
| Correct mass calculation with unit conversion | 6 |
| Correct kinetic energy and MT conversion | 6 |
| Correctly and honestly compares to the published range rather than overstating precision | 3 |
2. Olympus Mons and the Hawaiian Hotspot Track
Time = distance/speed = 2400.0 km / (0.07 km/yr) = 34.3 Myr ≈ 34 Myr, comparable in order of magnitude to Olympus Mons's estimated multi-hundred-Myr construction time. On Earth, the Pacific Plate's motion over the fixed Hawaiian hotspot spreads volcanic material into a long chain of progressively older, smaller volcanic islands and seamounts. On Mars, with no plate tectonics (Lecture 05), the lithosphere never moves relative to the underlying mantle hotspot, so all the erupted material from repeated eruptions over a comparably long timescale piles up in a single location, building one enormous volcano instead of a chain of smaller ones.
| Criterion | Points |
|---|
| Correct time calculation with unit conversion | 7 |
| Correctly explains the plate-tectonics/stationary-lithosphere contrast | 8 |
3. Crater Density and Venus’s Resurfacing Event
A surface that had simply never been struck by many impactors, or one that formed at the same time as the planet itself (4.5 Gyr ago) with no subsequent resurfacing, would both be expected to show a wide range of crater degradation states (heavily degraded ancient craters mixed with sharper younger ones accumulated steadily over 4.5 Gyr), because impacts would have been accumulating and being progressively degraded by subsequent smaller impacts and weathering throughout that entire span. Instead, Venus shows a low but narrow, roughly uniform range of crater degradation and density across its whole surface, which is the specific signature expected if nearly the entire surface was reset (buried by volcanism or otherwise resurfaced) at approximately the same time, comparatively recently, after which a much shorter interval of impact accumulation began anew.
| Criterion | Points |
|---|
| Correctly explains why continuous accumulation over 4.5 Gyr would look different | 8 |
| Correctly explains why a narrow degradation range specifically implies a discrete resurfacing event | 7 |
4. Simple versus Complex Crater Transition
The apparent contradiction is resolved because the simple-to-complex transition diameter depends on target material strength as well as gravity, and Mars's surface (a mix of regolith, sedimentary, and volcanic rock, in some places ice-rich) does not have uniformly stronger material strength than the Moon's regolith/rock in a way that offsets its stronger gravity in the direction assumed here; more importantly, the specific transition-diameter values quoted for the Moon versus Mars in the published literature reflect a real, measured combination of both the gravity dependence and material-strength differences (including, for icy or volatile-rich near-surface material on parts of Mars, weaker effective strength than typical dry lunar regolith), so gravity alone is not the only variable and should not be used in isolation to predict the transition diameter, which is exactly the caveat this problem is testing.
| Criterion | Points |
|---|
| Correctly identifies that gravity is not the sole controlling variable | 8 |
| Correctly explains the role of target material strength as a compensating factor | 7 |
5. Mercury’s Crater-Count Volcanic History
Mercury, the smallest of the terrestrial planets (aside from dwarf planets), has the largest surface-to-volume ratio and is therefore expected, by the 1/R cooling-rate scaling from Lecture 04, to cool and lose active volcanism earliest among the terrestrial planets, all else equal; Mercury's crater-count-derived cessation of volcanism by roughly 3.7-3.9 Gyr ago is directly consistent with this expectation. Independent supporting evidence includes Mercury's widespread lobate scarps (Lecture 05), interpreted as a record of global contraction as its large core cooled and shrank -- itself a direct geological signature of the same rapid interior cooling implied by the crater-count volcanic-cessation age.
| Criterion | Points |
|---|
| Correctly applies the surface-to-volume cooling argument to Mercury | 8 |
| Correctly cites lobate scarps as independent supporting evidence | 7 |
Grading Notes
Award full marks for correct method, units, and a numeric result consistent with the arithmetic shown (allow reasonable rounding). Verify that the student re-derives the number for this problem's specific inputs rather than quoting a lecture example without adaptation. Watch specifically for unit-conversion errors (AU/m, days or years/seconds, bar/Pa) and for comparative-magnitude statements stated in the wrong direction (e.g., a tidal acceleration or greenhouse excess described as a multiple when it is actually a small fraction, or vice versa) -- these are the most common sources of silent error in this course.