A spacecraft of mass 470 kg rests on the surface of an asteroid of radius 1400 m and mass 2.0*10^12. How much energy must be expended so that the spacecraft may rise to a height of 2800 m above the surface of the asteroid?
Thanks.
Why did the astronaut bring a ladder to the asteroid? Because he wanted to reach new heights and go beyond the moon-walking! Now, let's calculate the energy required for the spacecraft to rise.
To calculate the energy, we can use the formula:
E = m * g * h
Where:
E is the energy,
m is the mass,
g is the gravitational acceleration, and
h is the height.
First, we need to find the gravitational acceleration on the surface of the asteroid. Using Newton's Law of Universal Gravitation:
F = (G * M * m) / r^2
Where:
F is the force of gravity,
G is the gravitational constant,
M is the mass of the asteroid, and
r is the radius of the asteroid.
Calculating the force of gravity:
F = (6.67 * 10^-11) * (2.0 * 10^12) * (470) / (1400)^2
Now, let's find the gravitational acceleration:
g = F / m
Finally, we can calculate the total energy:
E = (470) * (g) * (2800 - 1400)
Now, grab your calculator and let's get crunching those numbers!
To calculate the energy required to raise the spacecraft to a height above the surface of the asteroid, we need to consider the gravitational potential energy.
Gravitational potential energy can be calculated using the formula:
Potential energy = mass * gravity * height
Where:
- mass is the mass of the spacecraft,
- gravity is the gravitational acceleration (which we'll assume is approximately 9.8 m/s^2 on the asteroid's surface),
- height is the height above the surface.
Let's calculate:
mass = 470 kg
gravity = 9.8 m/s^2
height = 2800 m
Potential energy = 470 kg * 9.8 m/s^2 * 2800 m
Potential energy = 12,920,800 Joules
Therefore, the spacecraft must expend approximately 12,920,800 Joules of energy to rise to a height of 2800 m above the surface of the asteroid.
To find the amount of energy that needs to be expended for the spacecraft to rise to a height of 2800 m above the surface of the asteroid, we can use the concept of gravitational potential energy.
The formula for gravitational potential energy is given by:
Potential Energy = mass * gravity * height
where mass is the mass of the spacecraft, gravity is the gravitational acceleration, and height is the change in height.
First, we need to find the gravitational acceleration on the surface of the asteroid. The formula for gravitational acceleration is given by:
acceleration = (G * Mass of the asteroid) / (Radius of the asteroid)^2
where G is the gravitational constant (approximately 6.67430 x 10^-11 m^3 / kg / s^2).
Substituting the given values, we have:
acceleration = (6.67430 x 10^-11 * (2.0 * 10^12)) / (1400)^2
Calculating this expression, we find:
acceleration ≈ 1.02 m/s^2
Now, we can calculate the potential energy using the formula:
Potential Energy = mass * gravity * height
Substituting the given values, we have:
Potential Energy = (470 kg) * (1.02 m/s^2) * (2800 m)
Calculating this expression, we find:
Potential Energy ≈ 1.1967 x 10^6 Joules
Therefore, approximately 1.1967 x 10^6 Joules of energy must be expended for the spacecraft to rise to a height of 2800 m above the surface of the asteroid.