A2 Daily A Level Physics question
Two identical satellites orbit the same planet. Satellite A moves on a circular orbit of radius R. Satellite B follows an ellipse with closest distance R and farthest distance 4R. Assume only the planet’s gravity acts. Which statement must be true?
Answer
The correct answer is A.
Correct: A — Satellite B takes about 4 times longer per orbit than A, and at 4R its speed is one quarter of its speed at R. For B the semi-major axis is a = (R + 4R)/2 = 2.5R, so T_B/T_A = (a/R)^(3/2) = 2.5^(3/2) ≈ 2.5 × √2.5 ≈ 3.95 ≈ 4; along B’s path, equal areas in equal times imply r v_tan is constant, so v(4R) = (R/4R) v(R) = v(R)/4. As a check, in the circular limit r_ap → r_per the speed ratio tends to 1, consistent with constant speed. A is the only option consistent with period set by semi‑major axis and the equal‑area argument for the speed ratio. B uses the farthest distance (4R) as if it set the period, giving a factor 8 that would only follow if a = 4R; the speed ratio part happens to match but the period is wrong overall. C keeps the correct period factor but halves the apoapsis speed, confusing the inverse‑radius relation (r v constant) with a square‑root law from circular orbits. D claims equal periods because both reach R, but period is set by the semi‑major axis (and the longer, wider orbit of B), not by the minimum radius; the speed ratio part is otherwise correct.