Suppose a planet is discovered orbiting a distant star with 13 times the mass of the Earth and 1/13 its radius. How does the escape speed on this planet compare with that of the Earth?

Ve new
------- =
ve earth

Escape velocity derives from Ve = sqrt[2µ/r] where Ve = escape velocity in ft/sec., µ = the gravitational constant of the earth and r = the surface radius.

µ can also be stated as GM where G = the gravitational and M = the mass of the planet.

Therefore, Ve can be simply expressed by
Ve = sqrt[2(13)µ/(r/13)] = sqrt[169(2µ/r] =
13 times earth's escape velocity of Ve = 36,747fps = 25,054mph or 325,712mph.

V(new)/V(earth) = 13

Well, let's do some math here. Since we're talking about escape speed, it's all about the ratio of gravitational potential energy to kinetic energy. Now, the escape speed on Earth (ve earth) is about 11.2 km/s. So, let's plug in the numbers we have for this distant planet into our equation and see what we get:

Ve new / ve earth = sqrt((2GMnew) / (2GMearth)) / sqrt(Rnew / Rearth)

Now, since the mass of the planet is 13 times that of Earth, and the radius is 1/13 of Earth's radius, we can simplify this equation a bit:

Ve new / ve earth = sqrt(13 * (1/13)) / sqrt((1/13) / 1)

And after doing some more calculations, I've got the answer for you! The escape speed on this distant planet is the same as on Earth. So don't worry, you won't need any more rocket fuel to get off that planet than you would on Earth. Just be careful not to accidentally book a one-way ticket!

To compare the escape speeds of two planets, we can use the equation:

Vescape = √(2GM/r)

Where:
- Vescape is the escape speed
- G is the gravitational constant (approximately 6.67 x 10^-11 N(m/kg)^2)
- M is the mass of the planet
- r is the radius of the planet

We are given that:

Mass of the new planet (Mnew) = 13 times the mass of Earth (Mearth)
Radius of the new planet (rnew) = 1/13 times the radius of Earth (rearth)

Using this information, we have:

Mnew = 13 * Mearth
rnew = (1/13) * rearth

Now, we can compare the escape speeds:

Vescape, new / Vescape, earth = √(2GMnew/rnew) / √(2GMearth/rearth)

Substituting the values, we get:

Vescape, new / Vescape, earth = √[(2G * 13Mearth) / ((1/13) * rearth)]

Simplifying further:

Vescape, new / Vescape, earth = √[(26G * Mearth) / rearth^2]

Since the expression for escape speed has the square of the radius in the denominator, we can simplify further:

Vescape, new / Vescape, earth = √[(26G * Mearth) / (rnew^2)]

Now, we can substitute the given values for the mass and radius:

Vescape, new / Vescape, earth = √[(26G * Mearth) / ((1/13 * rearth)^2)]

Vescape, new / Vescape, earth = √[(26G * Mearth) / (1/13)^2 * rearth^2)]

Simplifying further:

Vescape, new / Vescape, earth = √[(26G * Mearth) / (1/169) * rearth^2]

Vescape, new / Vescape, earth = √[169 * (26G * Mearth) / (rearth^2)]

Vescape, new / Vescape, earth = √[169 * (26G * Mearth / rearth^2)]

Since (26G * Mearth / rearth^2) is a constant, the equation becomes:

Vescape, new / Vescape, earth = √(169 * constant)

Vescape, new / Vescape, earth = √169

Vescape, new / Vescape, earth = 13

So, the escape speed on this planet is 13 times the escape speed on Earth.

To compare the escape speed of the new planet with that of Earth, we can use the concept of kinetic energy. The expression for escape velocity is derived from the conservation of energy. The formula is as follows:

Ve = sqrt(2 * G * Me / Re)

where:
Ve is the escape speed of the new planet,
G is the universal gravitational constant (approximately 6.67430 × 10^-11 m^3 kg^-1 s^-2),
Me is the mass of the new planet (13 times the mass of Earth),
and Re is the radius of the new planet (1/13 the radius of Earth).

Ve = sqrt(2 * G * (13 * Mearth) / (1/13 * Rearth))

Now, we can simplify this expression. Since we're given that the mass of the new planet is 13 times and the radius is 1/13th of that of Earth:

Ve = sqrt(2 * G * (13 * Mearth) / (1/13 * Rearth))
Ve = sqrt(2 * G * (13 * (Me / 13)) / (1/13 * (Re / 13)))
Ve = sqrt(2 * G * (Me) / (Re / 169))

Taking the ratio of the escape speeds of the new planet (Ve) and Earth (Ve_earth):

Ve_new / Ve_earth = (sqrt(2 * G * (Me) / (Re / 169))) / (sqrt(2 * G * (Mearth) / Rearth))

Simplifying that equation:

Ve_new / Ve_earth = sqrt(2 * G * (Me) / (Re / 169)) * sqrt(Rearth / (2 * G * Mearth))
Ve_new / Ve_earth = sqrt((2 * G * Me * Rearth) / (Re / 169)) / sqrt(2 * G * Mearth * Re)
Ve_new / Ve_earth = sqrt(13)

Therefore, the escape speed on this planet is √13 times the escape speed on Earth.