Escape speed from the Earth is close to 11·2 km s⁻¹. On another planet whose radius is half of the Earth's radius and whose mass density is four times that of the Earth, the escape speed in km s⁻¹ will be close to:
- (a)11·2
- (b)15·8
- (c)5·6
- (d)7·9
Correct — A, 11·2 km s⁻¹. Escape speed is v = sqrt(2GM/R). Replacing mass M by density x volume, M = rho x (4/3)piR^3, gives v = sqrt((8piG/3) x rho x R^2), so v is proportional to R x sqrt(rho). For the new planet R is halved and rho is four times larger, making the ratio (1/2) x sqrt(4) = (1/2) x 2 = 1. The escape speed therefore stays essentially the same as Earth's — about 11.2 km/s.
- (b)15·8 — About 11.2 x sqrt(2). This comes from multiplying by sqrt(density ratio) = sqrt(4) = 2 but forgetting the radius factor of 1/2 — using only the density change.
- (c)5·6 — Exactly half of 11.2. This results from applying only the radius factor (R halved) and ignoring the fourfold density increase.
- (d)7·9 — About 11.2/sqrt(2), the value you would get from a mistaken combination of the factors; it does not follow from v proportional to R x sqrt(rho), which gives exactly 11.2.
Escape speed is the minimum launch speed that lets an object leave a planet's gravity without further propulsion. From energy conservation it equals sqrt(2GM/R), where M is the planet's mass and R its radius. It does not depend on the mass of the escaping object.
When a problem gives density rather than mass, rewrite M as (density) x (volume) = rho x (4/3)piR^3. Substituting reduces the escape speed to v proportional to R x sqrt(rho), which turns the question into simple scaling. Track the radius and density factors separately, and the two changes here cancel exactly.
- Escape speed v = sqrt(2GM/R); Earth's value is about 11.2 km/s.
- Writing mass as density x volume gives v proportional to R x sqrt(density).
- Escape speed is independent of the escaping body's mass.
- For this planet: (R halved) x sqrt(density x4) = (1/2) x 2 = 1, so v is unchanged.
- Applying only the density change or only the radius change instead of both together.
- Confusing escape speed with orbital speed, which differ by a factor of sqrt(2).
Asked as a scaling problem — compare escape speed, g, or orbital speed on a planet whose radius and density differ from Earth's.
Assertion (A): Artificial satellites are always launched from the earth in the eastward direction. Reason (R): The earth rotates from west to east and so the satellite attains the escape velocity.
- (a) Both A and R are individually true and R is the correct explanation of A
- (b) Both A and R are individually true but R is not a correct explanation of A
- (c) A is true but R is false
- (d) A is false but R is true
Answer(c) A is true but R is false
Same concept — escape velocity. UPSC's reason is marked false because a satellite needs only orbital velocity, not escape velocity, sharpening the escape-vs-orbital distinction that underlies this NDA scaling problem.
- practice — not a real PYQ
Escape speed from a planet depends on which of the following?
- (a)Mass of the escaping object
- (b)The planet's mass and radius
- (c)The direction of launch only
- (d)The colour of the object
Answer(b) The planet's mass and radius — v = sqrt(2GM/R), independent of the escaping object's mass.
- practice — not a real PYQ
If a planet had the same density as Earth but twice its radius, its escape speed would be:
- (a)the same
- (b)half
- (c)twice
- (d)four times
Answer(c) twice — since v is proportional to R x sqrt(density), doubling R at fixed density doubles the escape speed.