Final answer:
Planet B must have twice the mass of Planet A (mb = 2ma) for its gravitational acceleration (g) to be half of that on Planet A, assuming both planets have the same density.
Step-by-step explanation:
To find what mass Planet B (mb) must have so that its gravitational acceleration (g) is half that of Planet A, we use Newton's law of universal gravitation. Gravitational acceleration on the surface of a planet is given by the formula g = Gm/r², where G is the gravitational constant, m is the mass of the planet, and r is the radius of the planet.
Since the radius of Planet A is half that of Planet B, we can denote the radius of Planet A as r and that of Planet B as 2r. If the gravitational acceleration on B is half that on A, we have gB = 1/2gA. Substituting this into the equations for gravitational acceleration, we get:
gA = Gma/r²
gB = Gmb/(2r)² = Gmb/4r²
From gB = 1/2gA, we can derive:
Gmb/4r² = 1/2(Gma/r²)
Solving for mb, we find:
mb = 2ma
Therefore, the mass of Planet B must be twice the mass of Planet A for gravitational acceleration on B to be half of that on A, assuming their densities are homogeneous.