The Question is from neet-physics gravitation , and it was created by the author P.NARASIMHA

Question

A satellite is in a low, circular orbit around Earth.  A small, impulsive retro-rocket burn is executed, reducing the satellite's speed by a small amount $\Delta v$. Which of the following best describes the resulting orbit immediately after the burn?

P.NARASIMHA · Accepted Answer

## CORE INFORMATION
✓ **Answer**: Option D
## CONCEPT FRAMEWORK
**Primary Concept**: Orbital Mechanics and Energy Conservation
**Related Topics**:  Elliptical Orbits, Orbital Velocity, Conservation of Angular Momentum
**Prerequisites**: Understanding of orbital motion, the concept of a retro-rocket burn, and the shapes of orbits.
## SOLUTION PATHWAY
**Given Information**:
*   Initial orbit: Circular
*   Action: Retro-rocket burn (reduction in speed $\Delta v$)
*   Burn is impulsive (occurs over a very short period of time).
*   The burn is in the direction opposite to the velocity.
**Method & Steps**:
1.  **Initial State**: The satellite is initially in a circular orbit, meaning its velocity is perpendicular to the radius vector from Earth, and its kinetic and potential energy are balanced for a circular path at that radius.
     Reasoning: Circular orbits require a specific balance between kinetic and potential energy.
     Formula/Rule: Circular orbit velocity, $v = \sqrt{\frac{GM}{r}}$, where $G$ is the gravitational constant, $M$ is the Earth's mass, and $r$ is the orbital radius.
2.  **Retro-Rocket Burn Effect**: The retro-rocket burn reduces the satellite's velocity at the point of the burn. This means at this point the satellite has less kinetic energy than required for the circular orbit.
    Reasoning: Reducing the velocity lowers the satellite's kinetic energy.
    Key Point: The burn happens very quickly, and we can consider it instantaneous.
3.  **Resulting Orbit**: When the speed is reduced, the satellite's new orbit will be elliptical. Because the velocity at the burn point is now less than required for a circular orbit at that radius, the satellite will fall to a lower altitude. The point where the burn occurs becomes the perigee (closest point to Earth) of the ellipse. The opposite point on the orbit will be the apogee (farthest point from Earth).
    Result: An elliptical orbit with the perigee at the burn point and the apogee diametrically opposite.
    Verification: This aligns with the behaviour of orbits when velocity is changed.
## OPTION ANALYSIS
[A] :x: A smaller, circular orbit. This would require a continuous reduction in speed, not an impulsive one.
[B] :x: A larger, circular orbit. This would require an increase in speed, not a decrease.
[C] :x: An elliptical orbit with the apogee at the burn point and the perigee diametrically opposite. The burn point is where the satellite's velocity is reduced, which makes it the perigee.
[D] ✓: An elliptical orbit with the perigee at the burn point and the apogee diametrically opposite. This is the correct description of the resulting orbit after a retro-rocket burn.
## LEARNING AIDS
**Common Mistakes**:
1.  **Confusing Perigee and Apogee**:  Remember that the perigee is the point where the satellite is closest to Earth and has its highest speed, while the apogee is the point farthest from Earth and has its lowest speed.
2.  **Thinking speed reduction results in a larger orbit**: A reduction in speed at a point in the circular orbit makes the satellite fall towards Earth. This means the new orbit must have a smaller radius at the burn point, making it the perigee.
**Quick Tips**:
1.  **Visualizing the burn**: Imagine the satellite being 'pulled' inwards at the point of the burn due to reduced velocity. This helps to visualize the perigee.
2.  **Energy**:  Reducing speed decreases the kinetic energy, and hence the total energy of the orbit, which means the resulting orbit must be smaller in some way compared to the original orbit.

P.NARASIMHA · Answer

A smaller, circular orbit.

P.NARASIMHA · Answer

A larger, circular orbit.

P.NARASIMHA · Answer

An elliptical orbit with the apogee at the burn point and the perigee diametrically opposite.

P.NARASIMHA · Answer

The correct Option is "An elliptical orbit with the perigee at the burn point and the apogee diametrically opposite."

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