A distant planet has a mass of 0.6400 ME and a radius of 0.9000 RE. What is the ratio of the escape speed from this planet to the escape speed from the earth?

To find the ratio of the escape speed from the distant planet to the escape speed from Earth, we can use the formula for escape speed:

Escape Speed = √(2 * gravitational constant * mass of the planet / radius of the planet)

The escape speed from Earth, let's call it vE, can be calculated using the same formula, but with the mass and radius of Earth.

Let me calculate both escape speeds and then we can find their ratio.

For the distant planet:
Mass of the planet (M) = 0.6400 times the mass of Earth (ME)
Radius of the planet (R) = 0.9000 times the radius of Earth (RE)

Now, let's calculate the escape speed from the planet (vP):

vP = √(2 * gravitational constant * M / R)

For Earth:
Mass of Earth (ME) = 5.972 × 10^24 kilograms
Radius of Earth (RE) = 6,371 kilometers

Let's calculate the escape speed from Earth (vE):

vE = √(2 * gravitational constant * ME / RE)

Now that we have both escape speeds, we can find their ratio:

Ratio = vP / vE

Let me calculate these values for you.