COMEDK 2024 Shift 1 Paper

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=√‌

2GM

R

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 ‌‌=9MX
R‌Earth ‌‌=4RX
The escape velocity for planet X is:
ve,X=√‌

2GMX

RX

The escape velocity for Earth is:

ve,Earth=√‌

2GM‌Earth ‌

R‌Earth ‌

Substitute M‌Earth ‌=9MX and R‌Earth ‌=4RX :
However, note that:
ve,X=√‌

2GMX

RX

So, the ratio of the escape velocity of planet X to Earth is:
‌

ve,X

ve,‌ Earth ‌

=‌

√ 2GMX RX

3√‌ GMX 2RX

=‌

√ 2GMX RX

3⋅√‌ GMX 2RX

‌=‌

1

3

⋅√‌

2GMX

RX

⋅‌

2RX

GMX

‌=‌

1

3

√4
‌=‌

1

3

⋅2
‌=‌

2

3

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