To find the ratio of the maximum speed required by a rocket to escape the gravitational force of planet X to that of Earth, we must consider the escape velocity formula:
ve=√‌where:
ve is the escape velocity
G is the gravitational constant
M is the mass of the planet
R is the radius of the planet
Given:
The mass of Earth (
M‌Earth ‌) is nine times the mass of planet X
(MX), The radius of Earth (
R‌Earth ‌ ) is four times the radius of planet
X(RX).
So, we have:
M‌Earth ‌‌=9MXR‌Earth ‌‌=4RX The escape velocity for planet X is:
ve,X=√‌The escape velocity for Earth is:
ve,Earth=√‌| 2GM‌Earth ‌ |
| R‌Earth ‌ |
Substitute
M‌Earth ‌=9MX and
R‌Earth ‌=4RX :
However, note that:
ve,X=√‌ So, the ratio of the escape velocity of planet X to Earth is:
‌=‌ =‌ ‌=‌⋅√‌⋅‌‌=‌√4‌=‌⋅2‌=‌ Therefore, the ratio of the maximum speed required by a rocket to escape the gravitational force of planet X to that of Earth is:
2∕3