1. Hubble's law prediction
v = H₀d = 71.5 × 250 = 17879 km/s.
| Criterion | Points |
|---|---|
| Correct formula application | 8 |
| Correct final velocity | 7 |
Worked Solutions
2. Hubble tension range
t(H₀=73) = 13.39 Gyr; t(H₀=67) = 14.59 Gyr. The lower H₀ (67 km/s/Mpc) gives the larger (older) Hubble time, since t ∝ 1/H₀.
| Criterion | Points |
|---|---|
| Correct Hubble time for both endpoints | 8 |
| Correct identification of which is larger and why (inverse relationship) | 7 |
3. Cluster velocity dispersion and dark matter
Just as M(R) = V²R/G for a rotating galaxy, a cluster's dynamical mass can be estimated from its member galaxies' velocity dispersion and size using the same virial-type scaling: large velocities require large gravitating mass to keep the cluster bound, and the implied mass from 1,000 km/s velocities vastly exceeds the visible galaxy light. Independent supporting evidence: gravitational lensing of background sources by the cluster directly maps its total mass distribution and consistently shows far more mass than the visible galaxies account for (e.g., the Bullet Cluster).
| Criterion | Points |
|---|---|
| Correct virial-type mass reasoning | 8 |
| Correct independent evidence citation (lensing) | 7 |
4. Textbook problem: the cosmic microwave background
Full-credit answers should explain: the Big Bang model specifically predicted a nearly uniform, blackbody radiation field left over from the hot, dense early universe, redshifted by cosmic expansion to microwave wavelengths (a few Kelvin) today. The Steady State model, which posited a universe with no beginning and continuous matter creation, made no such prediction and had no natural explanation for a uniform microwave background. Penzias and Wilson's accidental detection of exactly this radiation field, with the predicted near-perfect blackbody spectrum and near-perfect isotropy, was therefore strong, specific confirmation of the Big Bang over its main historical competitor.
| Criterion | Points |
|---|---|
| Correct explanation of what the CMB is and why the Big Bang predicted it | 8 |
| Correct contrast with the Steady State model’s lack of prediction | 7 |
Grading Notes
Award full marks for correct method, units, and a numeric result consistent with the arithmetic shown (allow reasonable rounding). Verify the student re-derives the number rather than quoting it from the lecture without adapting it to the problem's specific inputs.