The Question is from neet-physics gravitation orbital-velocity, and it was created by the author VENKATESWARARAO

Question

A satellite is orbiting Earth in a highly elliptical orbit. At its closest approach (periapsis), its speed is $v_p$ and its distance from Earth's center is $r_p$. At its farthest point (apoapsis), its distance is $r_a$. If $r_a = 4r_p$, what is the satellite's speed at apoapsis?

VENKATESWARARAO · Accepted Answer

## CORE INFORMATION
✓ **Answer**: [B]
## CONCEPT FRAMEWORK
	**Primary Concept**: Conservation of Angular Momentum
	**Related Topics**: Orbital Mechanics, Kepler's Second Law
	**Prerequisites**: Understanding of angular momentum, elliptical orbits
## SOLUTION PATHWAY
**Given Information**:
	* Speed at periapsis: $v_p$
	* Distance at periapsis: $r_p$
	* Distance at apoapsis: $r_a$
	* Relationship: $r_a = 4r_p$
	* Find: Speed at apoapsis, $v_a$
**Method & Steps**:
	1. Apply the principle of conservation of angular momentum.
		Reasoning: In the absence of external torques, the angular momentum of the satellite is constant throughout its orbit. Angular momentum $L = I\omega = mvr$ for a point mass, where $m$ is mass, $v$ is velocity, and $r$ is the distance from center.
		Formula/Rule: $L_p = L_a$ or $m v_p r_p = m v_a r_a$
	2. Substitute the given relationship $r_a = 4r_p$ into the angular momentum conservation equation.
		Working: $m v_p r_p = m v_a (4r_p)$. Mass $m$ cancels out from both side, we get $v_p r_p = 4 v_a r_p$. Dividing both side by $r_p$, we get $v_p = 4v_a$
		Key Point: The mass of the satellite is constant, and angular momentum is conserved.
	3.  Solve for $v_a$.
		Result: $v_a = \frac{v_p}{4}$
		Verification:  The speed at apoapsis should be slower because it is farther from Earth, consistent with conservation of angular momentum.
 ## OPTION ANALYSIS
[A] :x: Incorrect, this would be the speed if $r_a = 2r_p$
[B] ✓: Correct, angular momentum is conserved, so if the radius quadruples, the velocity is reduced by a factor of four
[C] :x: Incorrect, speed at apoapsis should be less than speed at periapsis 
[D] :x: Incorrect, speed should be less than at periapsis
 ## LEARNING AIDS
	**Common Mistakes**:
		1. [Error 1]: Incorrectly applying the conservation of energy instead of angular momentum. [How to avoid]: Remember that energy conservation includes potential energy, which is more complex. Angular momentum conservation is simpler in this case since we are only looking at two points.
		2. [Error 2]: Forgetting that the distance at apoapsis is 4 times that at periapsis. [How to avoid]: Pay careful attention to the problem statement and double check the given relationships.
	**Quick Tips**:
		1. [Helpful shortcut]:  The ratio of velocities is inversely proportional to the ratio of distances when angular momentum is conserved.
		2. [Memory aid]: Think of a figure skater, they spin faster when their arms are closer to their body, similar to how a satellite moves faster at periapsis.

VENKATESWARARAO · Answer

$v_p/2$

VENKATESWARARAO · Answer

The correct Option is "$v_p/4$"

VENKATESWARARAO · Answer

$2v_p$

VENKATESWARARAO · Answer

$4v_p$

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