A satellite orbits around the earth in a circular orbit with a speed `upsilon` and orbital radius `r`. If it loses some energy, then `upsilon` and `r` chsnges as

Options

A. `upsilon` decreases and `r` increases
B. both `upsilon` and `r` decreases
C. `upsilon` increases and `r` decreases
D. both `upsilon` and `r` increases

Explanation

To analyze how the speed \( \upsilon \) and the orbital radius \( r \) of a satellite change when it loses energy, we can follow these steps:

### Step 1: Understand the relationship between speed, gravitational constant, mass of the Earth, and radius
The speed \( \upsilon \) of a satellite in a circular orbit is given by the formula:
\[ \upsilon = \sqrt{\frac{GM}{r}} \]
where \( G \) is the gravitational constant and \( M \) is the mass of the Earth.

### Step 2: Understand the total mechanical energy of the satellite
The total mechanical energy \( E \) of a satellite in orbit is the sum of its kinetic energy \( K \) and potential energy \( U \):
\[ E = K + U \]
The kinetic energy \( K \) is given by:
\[ K = \frac{1}{2} m \upsilon^2 \]
And the potential energy \( U \) is given by:
\[ U = -\frac{GMm}{r} \]
Thus, the total energy can be expressed as:
\[ E = \frac{1}{2} m \upsilon^2 - \frac{GMm}{r} \]

### Step 3: Analyze the effect of losing energy
When the satellite loses energy, its total mechanical energy \( E \) decreases. Since the total energy in a circular orbit is negative (as it is bound), a decrease in energy means that the satellite is moving to a lower energy state.

### Step 4: Determine how speed and radius change
From the formula for speed, we see that if the radius \( r \) decreases, the speed \( \upsilon \) must increase to maintain the balance of forces (gravitational force providing the necessary centripetal force).

Conversely, if the satellite loses energy and moves to a lower orbit (which corresponds to a smaller radius), the speed must increase to compensate for the decrease in radius.

### Conclusion
Thus, when a satellite loses energy:
- The orbital radius \( r \) **decreases**.
- The orbital speed \( \upsilon \) **increases**.