Question

A 55,000 kg space capsule is in a 28,000-km-diameter circular orbit around the moon. A brief but intense firing of its engine in the forward direction suddenly decreases its speed by 50%. This causes the space capsule to go into an elliptical orbit. What are the space capsule’s (a) maximum and (b) minimum distances from the center of the moon in its new orbit? Hint: You will need to use two conservation laws.

Accepted Answer

Step 1: Identify the given values and understand the problem. The mass of the space capsule is 55,000 kg, the initial orbit is circular with a diameter of 28,000 km (radius = 14,000 km), and the speed decreases by 50%. The problem involves transitioning from a circular orbit to an elliptical orbit, requiring the use of conservation of energy and conservation of angular momentum. Step 2: Apply the conservation of angular momentum. Angular momentum is conserved because no external torque acts on the system. The initial angular momentum in the circular orbit is given by $$ L_{initial} = m v_{initial} r $$, where $$ m  is $$the mass, $$ v_{initial}  is $$the initial orbital speed, and $$ r  is $$the radius of the circular orbit. After the speed decreases, the angular momentum is $$ L_{final} = m v_{final} r_{periapsis} . $$Set $$ L_{initial} = L_{final}  to $$relate the periapsis distance $$ r_{periapsis}  to $$the initial conditions. Step 3: Use the conservation of mechanical energy. The total mechanical energy in the orbit is the sum of kinetic energy and gravitational potential energy. For the initial circular orbit, $$ E_{initial} = \frac{1}{2} m v_{initial}^2 - \frac{G M m}{r} $$, where $$ G  is $$the gravitational constant, $$ M  is $$the mass of the moon, and $$ r  is $$the radius of the circular orbit. For the elliptical orbit, $$ E_{final} = \frac{1}{2} m v_{final}^2 - \frac{G M m}{r_{periapsis}} . $$Set $$ E_{initial} = E_{final}  to $$find the relationship between the periapsis and apoapsis distances. Step 4: Relate the apoapsis and periapsis distances to the semi-major axis of the elliptical orbit. The semi-major axis $$ a  is $$given by $$ a = \frac{r_{periapsis} + r_{apoapsis}}{2} . $$Use the energy and angular momentum equations to solve for $$ r_{periapsis} $$ and $$ r_{apoapsis} . $$The apoapsis distance $$ r_{apoapsis}  is $$the farthest point from the moon, and the periapsis distance $$ r_{periapsis}  is $$the closest point. Step 5: Solve the equations step by step. First, calculate the initial orbital speed $$ v_{initial} $$ using $$ v_{initial} = \sqrt{\frac{G M}{r}} . $$Then, find $$ v_{final} = 0.5 v_{initial} . $$Substitute $$ v_{initial} $$ and $$ v_{final} $$ into the angular momentum and energy conservation equations to solve for $$ r_{periapsis} $$ and $$ r_{apoapsis} . $$These distances represent the maximum and minimum distances from the center of the moon in the new elliptical orbit.

Verified video answer for a similar problem:

Key Concepts

Conservation of Energy

Conservation of Angular Momentum

Elliptical Orbits