Final answer:
The ratio of the accelerations due to gravity of the two planets is 9:14.
Step-by-step explanation:
To find the ratio of the accelerations due to gravity of two planets, we need to use Newton's Law of Universal Gravitation. The law states that the gravitational force between two objects is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centers. This can be expressed as:
F = G * (m1 * m2) / r^2
where F is the gravitational force, G is the gravitational constant, m1 and m2 are the masses of the two objects, and r is the distance between their centers.
In this case, the ratio of masses is given as 2:3 and the ratio of radii is given as 4:7. We can assume that the masses and radii are directly proportional, so we can write:
m1/m2 = 2/3 and r1/r2 = 4/7
From the ratio of radii, we can also assume that the ratio of distances from the planets to their centers is the same as the ratio of radii. Therefore, we can write:
d1/d2 = 4/7
Now, let's substitute the values into the equation for the gravitational force:
F1 = G * (m1 * m2) / (r1^2)
F2 = G * (m1 * m2) / (r2^2)
Dividing the two equations, we get:
F1/F2 = G * (m1 * m2) / (r1^2) / (G * (m1 * m2) / (r2^2))
Canceling out the common terms, we get:
F1/F2 = r2^2 / r1^2
Substituting the values for the ratios of radii, we get:
F1/F2 = (4/7)^2 / (2/3)^2
Simplifying the expression, we get:
F1/F2 = (16/49) / (4/9) = 16/49 * 9/4 = 144/196 = 9/14
Therefore, the ratio of the accelerations due to gravity is 9:14.