Find the minimum kinetic energy needed for a 3.7×104-kg rocket to escape the Moon.
G*M(rocket)*M(moon)/R(moon)
where G is the universal gravity constant.
It can also be written
M(rocket)*g'(moon)*R(moon)
where g' is the acceleration of gravity at the Moon's surface. It is about g/6, or 1.6 m/s^2.
Well, to answer your question, the minimum kinetic energy needed for a rocket to escape the Moon is directly proportional to the amount of boredom you can generate. So, to calculate that, we'll need to measure how many people fall asleep while reading about rocket science. Let's just say the minimum kinetic energy needed is inversely proportional to the number of awake individuals, which is hopefully higher than the minimum number of laughs you'll get from this answer.
To find the minimum kinetic energy needed for a rocket to escape the Moon, we can use the formula for gravitational potential energy and equate it to the initial kinetic energy.
The gravitational potential energy is given by the formula:
PE = – (G * m1 * m2) / r
Where:
PE = Gravitational potential energy
G = Gravitational constant = 6.67430 × 10-11 N m2/kg2
m1 = Mass of the Moon = 7.342 × 1022 kg
m2 = Mass of the rocket = 3.7 × 104 kg
r = Distance between the center of the Moon and the rocket = radius of the Moon = 1.737 × 106 m
By equating the gravitational potential energy to the initial kinetic energy (KE), we can write:
PE = KE
Solving for KE, we get:
KE = – (G * m1 * m2) / r
Plugging in the values, we have:
KE = - (6.67430 × 10-11 N m2/kg2 * 7.342 × 1022 kg * 3.7 × 104 kg) / (1.737 × 106 m)
Simplifying the equation, we get:
KE = - (2.42569 × 1028 N m/kg) / (1.737 × 106 m)
Calculating the value, we find:
KE ≈ - 1.398 × 1022 J
Note that the negative sign indicates that the kinetic energy required is negative, which implies that the rocket needs to gain this amount of kinetic energy to escape the gravitational pull of the Moon.
To find the minimum kinetic energy needed for a rocket to escape the Moon, we can use the concept of gravitational potential energy and the conservation of mechanical energy.
The equation for gravitational potential energy is:
Potential Energy = mgh
Where:
m = mass of the object (rocket)
g = acceleration due to gravity (on the surface of the Moon)
h = height above the reference point (usually defined as the Moon's surface)
The conservation of mechanical energy states that the total mechanical energy of a system remains constant. In this case, the mechanical energy is the sum of the kinetic energy and the potential energy of the rocket.
To escape the Moon, the rocket needs to reach a height where the potential energy is zero. This is at an infinite distance from the Moon's surface.
Therefore, the minimum kinetic energy needed for the rocket to escape the Moon is equal to the gravitational potential energy at the surface of the Moon. This can be calculated as:
Kinetic Energy = Potential Energy = mgh
Given:
mass of the rocket, m = 3.7 × 10^4 kg
We need to find the value of g and h to calculate the minimum kinetic energy.
The acceleration due to gravity on the surface of the Moon is approximately 1.6 m/s^2. This is about 1/6th of the acceleration due to gravity on Earth.
The height above the surface of the Moon, h, is zero because we are considering the surface as the reference point.
Substituting the values into the equation, we have:
Kinetic Energy = mgh = (3.7 × 10^4 kg) × (1.6 m/s^2) × (0 m)
= 0 Joules
Therefore, the minimum kinetic energy needed for a 3.7 × 10^4 kg rocket to escape the Moon is 0 Joules.