A2 Daily A Level Physics question
Two airless spherical planets are otherwise similar. Planet X has mass M and radius R. Planet Y has mass 2M and radius 2R. A small probe starts from rest very far away (where gravitational potential is taken as zero) and falls straight to each surface. Which statement must be true?
Answer
The correct answer is B.
Correct: B — At Y the surface gravitational field is half that at X, yet a fall from very far away gives the same impact speed on both planets. Surface g depends on mass divided by radius squared, so for Y it scales as (2M)/(2R)^2 = (1/2)(M/R^2); the impact speed from infinity depends on the drop in potential per kilogram, which scales as M/R, giving (2M)/(2R) = M/R for both, so speeds match (potential starts at 0 at infinity and is more negative at the surface). A The surface fields are not the same: doubling M and R together halves g; also the larger mass does not increase impact speed here because the increased radius cancels in the potential drop. B This is the only statement consistent with g ∝ M/R^2 giving 1/2 and the potential drop from infinity depending on M/R being equal for X and Y. C This reverses both scalings: g does not double (it halves), and the impact speed is not larger on Y because the potential drop per kilogram is the same. D The impact speeds are not reduced by 1/2; although g is halved, the potential drop from infinity is unchanged, so the speeds are equal, not scaled with g.