Final answer:
The escape velocity for the planet with four times the radius will be half of that of the smaller planet, as escape velocity is inversely proportional to the square root of the radius if the mass is constant.
Step-by-step explanation:
If two planets have the same mass, but one has four times the radius of the other, their escape velocities will differ significantly. The escape velocity of a planet is calculated using the formula v = sqrt(2GM/R), where G is the gravitational constant, M is the mass of the planet, and R is the radius of the planet. As the radius increases, the escape velocity decreases because the radius is in the denominator of the formula.
For the planet with four times the radius, the escape velocity will be half that of the smaller planet, since escape velocity is inversely proportional to the square root of the radius (if mass is constant). Hence, the larger radius reduces the escape velocity by a factor of √4, which is 2. This illustrates how significant changes in planetary radius can affect escape velocities, even when planetary mass remains unchanged.