Gravitation and potential energy

In summary, the conversation discusses a problem in which the gravitational potential energy due to the moon must be included when calculating the total gravitational potential energy of a spacecraft traveling from Earth to the moon. The problem involves finding the total gravitational potential energy of the particle-earth and particle-moon systems, the distance of a point where the net gravitational force is zero, the required launch speed for a spacecraft to reach this point, and the impact speed of a spacecraft launched from Earth's surface. The individual asking the question encountered difficulties in solving the problem on a physics platform called Mastering Physics.
  • #1
cuman12
3
0
In the example 12.5 (Section 12.3) in the textbook we ignored the gravitational effects of the moon on a spacecraft en route from the Earth to the moon. In fact, we must include the gravitational potential energy due to the moon as well. For this problem, you can ignore the motion of the Earth and moon.

A) If the moon of mass m(m) has radius R(M) and the distance between the centers of the Earth and the moon is R(EM), find the total gravitational potential energy of the particle-earth and particle-moon systems when a particle with mass m is between the Earth and the moon, and a distance r from the center of the earth. Take the gravitational potential energy to be zero when the objects are far from each other. Take the mass of Earth as m(E)

B) There is a point along a line between the Earth and the moon where the net gravitational force is zero. Use the expression derived in part (a) to find the distance of this point from the center of the earth.

C) With what speed must a spacecraft be launched from the surface of the Earth just barely to reach this point?

D) If a spacecraft were launched from the Earth's surface toward the moon with an initial speed of 11.2 {\rm km/s}, with what speed would it impact the moon?
 
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  • #2
Perhaps it would help if you would show where you encountered problems in your attempts to solve this problem?
 
  • #3
It was in mastering physics
 
  • #4
Try!
 
  • #5


A) The total gravitational potential energy of the particle-Earth system can be calculated using the equation U(E) = -Gm(m)m(E)/R(EM), where G is the gravitational constant. Similarly, the total gravitational potential energy of the particle-moon system can be calculated using the equation U(M) = -Gm(m)m(M)/R(EM) - Gm(m)m(E)/r. Therefore, the total gravitational potential energy of the particle-Earth and particle-moon systems will be U(T) = U(E) + U(M) = -Gm(m)m(E)/R(EM) - Gm(m)m(M)/R(EM) - Gm(m)m(E)/r.

B) To find the point where the net gravitational force is zero, we can set the total gravitational potential energy calculated in part (a) equal to zero and solve for r. This will give us the distance r from the center of the Earth where the net gravitational force is zero.

C) To reach the point where the net gravitational force is zero, a spacecraft must have enough kinetic energy to overcome the gravitational potential energy. This can be calculated using the equation KE = U(T), where KE is the kinetic energy and U(T) is the total gravitational potential energy calculated in part (a). Therefore, the speed required for the spacecraft to reach this point can be calculated using the equation v = sqrt(2U(T)/m), where m is the mass of the spacecraft.

D) If a spacecraft is launched from the Earth's surface with an initial speed of 11.2 km/s, it will have kinetic energy equal to 1/2mv^2. This kinetic energy must be equal to the total gravitational potential energy calculated in part (a) for the spacecraft to reach the point where the net gravitational force is zero. Therefore, we can set 1/2mv^2 = U(T) and solve for v to find the final speed of the spacecraft when it impacts the moon.
 

1. What is gravitation and how does it work?

Gravitation is the force of attraction between two objects with mass. This force is dependent on the mass of the objects and the distance between them. The larger the mass of an object, the greater its gravitational pull. This force also decreases as the distance between the objects increases.

2. How is potential energy related to gravitation?

Potential energy is the energy an object possesses due to its position or configuration. In the case of gravitation, potential energy is the energy that an object has due to its position in a gravitational field. The closer an object is to a larger mass, the more potential energy it has.

3. What is the formula for calculating gravitational potential energy?

The formula for gravitational potential energy is U = mgh, where U is the potential energy, m is the mass of the object, g is the acceleration due to gravity, and h is the height of the object above the ground or reference point.

4. How does potential energy change when an object is moved in a gravitational field?

When an object is moved in a gravitational field, its potential energy changes. If the object is moved closer to the source of the gravitational field, its potential energy increases. On the other hand, if the object is moved further away, its potential energy decreases.

5. What are some real-life examples of gravitation and potential energy in action?

There are many examples of gravitation and potential energy in our daily lives. Some examples include objects falling to the ground due to the Earth's gravitational pull, planets orbiting around the sun, and a roller coaster gaining potential energy at the top of a hill before converting it to kinetic energy during the descent.

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