Final answer:
The ratio of gravitational acceleration on planet X to that on Earth would be 0.16:1, using the formula derived from Newton's law of universal gravitation and considering that planet X has four times the mass and five times the diameter of Earth.
Step-by-step explanation:
The question deals with the gravitational acceleration of a hypothetical planet 'X' compared to Earth's gravitational acceleration. To find the ratio of gravitational acceleration gX to that of Earth's ge, we use Newton's law of universal gravitation, which states that the gravitational acceleration (g) on the surface of a spherical body like a planet is directly proportional to its mass (M) and inversely proportional to the square of its radius (r). The formula for gravitational acceleration is g = G * M / r2, where G is the gravitational constant.
Given the parameters for planet X, we have a mass 4 times that of Earth (4M) and a diameter 5 times Earth's (therefore, a radius 5 times, since diameter = 2 * radius). Substituting these into the formula, the acceleration due to gravity on planet X would be gX = G * (4M) / (5R)2 = G * 4M / 25R2 = (4/25) * G * M / R2. On Earth, the acceleration is ge = G * M / R2.
Dividing the acceleration of planet X by that of Earth, we get gX / ge = (4/25) * (G * M / R2) / (G * M / R2) = 4/25 = 0.16. Therefore, the ratio of gravitational acceleration on planet X to that on Earth would be 0.16:1.