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?

since v^2 = 2GM/r,

replacing with the new values gives a new v, with the ratio

vnew^2/vold^2 = 2G(.64M)/(.9r) / 2GM/r
= 0.711
So, vnew/vold = 0.84

To find the ratio of the escape speed from the distant planet to the escape speed from Earth, we need to calculate the escape speed for both planets.

The formula for escape speed is given by:

v = √(2GM/R)

Where:
- v is the escape speed
- G is the gravitational constant (approximately 6.67430 × 10^-11 m^3 kg^−1 s^−2)
- M is the mass of the planet
- R is the radius of the planet

We can now calculate the escape speed for the distant planet.

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

To calculate the escape speed from the distant planet (v_distant), we substitute the values into the formula:

v_distant = √(2G(0.6400 ME)/(0.9000 RE))

To calculate the escape speed from Earth (v_earth), we substitute the values into the formula:

v_earth = √(2G(ME)/RE)

Now, to find the ratio of the escape speeds, we divide the escape speed from the distant planet by the escape speed from Earth:

Ratio = v_distant / v_earth
= (√(2G(0.6400 ME)/(0.9000 RE))) / (√(2G(ME)/RE))

By simplifying the equation, you will get the final ratio of the escape speeds for the distant planet to Earth.