For two identical satellites in circular motion around the Earth, which statement is true?
a. The one in the higher orbit has more kinetic energy
b. The one in the lower orbit has more total energy
c. The one in the lower orbit has less total energy
d. Both have the same total energy
The correct answer is c. The one in the lower orbit has less total energy.
When an object is in circular motion, its kinetic energy and potential energy change. The kinetic energy of an object is given by the equation KE = 1/2 mv^2, where m is the mass of the object and v is its velocity. The potential energy of an object in circular motion is given by the equation PE = -GMm/r, where G is the gravitational constant, M is the mass of the central body (in this case, Earth), m is the mass of the satellite, and r is the distance between the satellite and the center of the Earth.
Since the two satellites are identical, they have the same mass. In circular motion, the velocity of an object depends on its distance from the center of rotation. The satellite in the higher orbit has a greater distance from the center compared to the satellite in the lower orbit. Therefore, it has a higher velocity and hence a higher kinetic energy. However, the potential energy of an object in circular motion is inversely proportional to its distance from the center. So, the one in the lower orbit has a smaller potential energy.
Total energy is the sum of kinetic energy and potential energy. Since the satellite in the lower orbit has less potential energy but the same mass as the satellite in the higher orbit, it would have less total energy. Thus, the correct statement is that the satellite in the lower orbit has less total energy (option c).
To determine which statement is true, we need to understand the relationship between the orbit of a satellite and its kinetic and total energy.
First, let's consider the basics of circular motion. When a satellite moves in a circular orbit around the Earth, it experiences a gravitational force pulling it towards the Earth's center. This gravitational force provides the centripetal force necessary for the satellite to stay in its orbit.
Now, let's analyze each statement:
a. The one in the higher orbit has more kinetic energy:
To determine the kinetic energy of an object, we can use the equation:
KE = (1/2)mv²,
where KE is the kinetic energy, m is the mass of the satellite, and v is its velocity. The velocity can be calculated using the formula:
v = √(GM/r),
where G is the gravitational constant, M is the mass of the Earth, and r is the radius of the satellite's orbit.
If we compare two identical satellites in different orbits, the one in the higher orbit will have a larger value of r, resulting in a lower velocity and therefore lower kinetic energy. So, this statement is false.
b. The one in the lower orbit has more total energy:
Total energy is the sum of kinetic energy and potential energy. For a satellite in orbit, potential energy is given by:
PE = -GMm/r,
where PE is the potential energy.
As we mentioned earlier, the satellite in the lower orbit has a smaller value of r. Since potential energy is inversely proportional to r, the satellite in the lower orbit will have a smaller value of potential energy. Therefore, this statement is false.
c. The one in the lower orbit has less total energy:
Based on our analysis from statement b, we can conclude that the satellite in the lower orbit has less total energy. This statement is true.
d. Both have the same total energy:
From our explanations above, we can see that this statement is false. The satellites in different orbits have different total energies.
In conclusion, the correct statement is:
c. The one in the lower orbit has less total energy.