A comet orbits the Sun in a highly elliptical orbit. Does the comet have a constant (a) linear speed (b) angular speed (c) angular momentum (d) kinetic energy (e) potential energy (f) total energy throughout its orbit? Neglect any mass loss of the comet when it comes very close to the Sun.

Options

A. (i),(ii),(iii)
B. (iii),(iv),(v)
C. (iii) and (vi)
D. (ii),(iii) and (vi)

Explanation

To determine whether the comet has constant values for linear speed, angular speed, angular momentum, kinetic energy, potential energy, and total energy throughout its elliptical orbit around the Sun, we can analyze each aspect step by step.

### Step-by-Step Solution:

1. **Linear Speed (a)**:
- In an elliptical orbit, the distance from the Sun (the focal point) changes as the comet moves along its path. According to Kepler's laws, the comet moves faster when it is closer to the Sun and slower when it is farther away.
- **Conclusion**: The linear speed of the comet is **not constant**.

2. **Angular Speed (b)**:
- Angular speed is defined as the rate of change of angular displacement. Since the linear speed is not constant and the radius (distance from the Sun) changes, the angular speed also changes.
- **Conclusion**: The angular speed of the comet is **not constant**.

3. **Angular Momentum (c)**:
- Angular momentum (L) is given by the formula \( L = mvr \), where \( m \) is the mass, \( v \) is the linear speed, and \( r \) is the radius. In the absence of external torque, angular momentum is conserved.
- Since there are no external torques acting on the comet, its angular momentum remains constant throughout its orbit.
- **Conclusion**: The angular momentum of the comet is **constant**.

4. **Kinetic Energy (d)**:
- Kinetic energy (KE) is given by the formula \( KE = \frac{1}{2}mv^2 \). Since the linear speed is not constant, the kinetic energy will also vary as the speed changes.
- **Conclusion**: The kinetic energy of the comet is **not constant**.

5. **Potential Energy (e)**:
- Gravitational potential energy (PE) is given by \( PE = -\frac{GMm}{r} \), where \( G \) is the gravitational constant, \( M \) is the mass of the Sun, \( m \) is the mass of the comet, and \( r \) is the distance from the Sun. As the comet moves closer or farther from the Sun, the potential energy changes.
- **Conclusion**: The potential energy of the comet is **not constant**.

6. **Total Energy (f)**:
- The total mechanical energy (E) of the comet is the sum of its kinetic and potential energy: \( E = KE + PE \). Although both kinetic and potential energy vary, the total energy remains constant throughout the orbit due to the conservation of mechanical energy in a closed system.
- **Conclusion**: The total energy of the comet is **constant**.

### Summary of Results:
- (a) Linear speed: **Not constant**
- (b) Angular speed: **Not constant**
- (c) Angular momentum: **Constant**
- (d) Kinetic energy: **Not constant**
- (e) Potential energy: **Not constant**
- (f) Total energy: **Constant**