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A scientist working for a space agency noticed that a Russian satellite of mass \(250 . \mathrm{kg}\) is on collision course with an American satellite of mass \(600 .\) kg orbiting at \(1000 . \mathrm{km}\) above the surface. Both satellites are moving in circular orbits but in opposite directions. If the two satellites collide and stick together, will they continue to orbit or crash to the Earth? Explain.

Short Answer

Expert verified
Answer: The satellites will crash to Earth.

Step by step solution

01

Find the velocity of each satellite before collision

First, we need to find the velocity of each satellite before collision. In order to do that, we use the formula for the orbital velocity, which is given by: \(v = \sqrt{\frac{GM}{r}}\) Where \(v\) is the orbital velocity, \(G\) is the gravitational constant, \(M\) is the mass of the Earth, and \(r\) is the distance from the center of the Earth to the satellite's orbit. Since both satellites are orbiting at the same altitude (\(1000 km\)), their radii are the same. The given altitude is above the surface. The Earth's radius (\(R_\oplus\)) is about \(6371\,\text{km}\). So the distance from the center to the orbit (r) is the sum of the Earth's radius and the altitude above the Earth's surface: \(r= R_\oplus +1000\, \text{km}= 6371\, \text{km} + 1000\, \text{km}= 7371\, \text{km}= 7.371 \times 10^6\, \text{m}\) Now, we can find the orbital velocity for each satellite: \(v = \sqrt{\frac{GM}{r}}\) Where \(GM = 3.986 \times 10^{14} \,\text{m}^3\text{s}^{-2}\) (the Earth's mass multiplied by the gravitational constant). \(v = \sqrt{\frac{3.986 \times 10^{14}\, \text{m}^3\text{s}^{-2}}{7.371 \times 10^6\, \text{m}}} = 7526.3\, \text{m/s}\) Both satellites have the same velocity, but they are moving in opposite directions.
02

Calculate the momentum of each satellite before collision

Since the satellites have opposite velocities, we can treat them as having velocities of \(7526.3\, \text{m/s}\) and \(-7526.3\, \text{m/s}\). Now, we can find the momentum for each satellite: \(p = mv\) For the Russian satellite: \(p_1 = (250\, \text{kg})(7526.3\, \text{m/s}) = 1.881575 \times 10^6\, \text{kg m/s}\) For the American satellite: \(p_2 = (600\, \text{kg})(-7526.3\, \text{m/s}) = -4.51578 \times 10^6\, \text{kg m/s}\)
03

Find the final velocity of the combined satellites after collision

Next, we can find the final velocity of the combined satellites after collision using conservation of momentum. The total momentum after the collision should be equal to the total momentum before the collision: \(p_{total} = p_1 + p_2 = 1.881575 \times 10^6\, \text{kg m/s} - 4.51578 \times 10^6\, \text{kg m/s}\) \(p_{total} = -2.634205 \times 10^6\, \text{kg m/s}\) Now, we can find the final velocity of the combined satellites: \(v_{final} = \frac{p_{total}}{m_{total}}\) Where \(m_{total}= 250\,\text{kg} + 600\,\text{kg}= 850\,\text{kg}\). \(v_{final} = \frac{-2.634205 \times 10^6\,\text{kg m/s}}{850\,\text{kg}} = -3099.064\,\text{m/s}\)
04

Compare the final velocity to the required velocity for stable orbit

Finally, we need to compare the final velocity with the required velocity for a stable orbit for the combined satellite. If the final velocity is greater than or equal to the required velocity, the combined satellite will continue to orbit; otherwise, it will crash to Earth. The required velocity for a stable orbit of an object with mass \(m_{total}\) at the same orbit is the same as the required velocity for either satellite: \(v_{required} = 7526.3\, \text{m/s}\) Since \(|v_{final}| = 3099.064\, \text{m/s} < v_{required} = 7526.3\, \text{m/s}\), we can conclude that the satellites will not continue to orbit and instead will crash to Earth.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Orbital Velocity
Understanding orbital velocity is crucial for grasping how objects remain in orbit around a planet. When a satellite orbits Earth, it must travel at a certain speed to counteract the planet's gravity pulling it downward. This speed is called orbital velocity.
Typically, the formula to calculate this velocity is: \[v = \sqrt{\frac{GM}{r}}\]where:
  • \(v\) is the orbital velocity,
  • \(G\) represents the gravitational constant \(6.674 \times 10^{-11} \, \text{m}^3\, \text{kg}^{-1} \text{s}^{-2}\),
  • \(M\) is the mass of the celestial body being orbited—in this case, Earth, which is approximately \(5.972 \times 10^{24}\) kg,
  • \(r\) is the radius of the orbit, the sum of the Earth’s radius and the satellite altitude above the surface.

For satellites at a height of \(1000\) km above the Earth's surface, the orbital velocity needed is approximately \(7526.3\) m/s. Achieving this ensures that they stay in a stable orbit without falling back to Earth.
Conservation of Momentum
The principle of conservation of momentum is fundamental in collision dynamics. It tells us that the total momentum of a closed system remains constant if no external forces are acting upon it.
Before the two satellites collide, each had a momentum product of its mass and velocity. Since they were moving in opposite directions:
The Russian satellite with a mass of \(250\) kg had a momentum of:\[p_1 = 250 \times 7526.3 \approx 1.881575 \times 10^6 \, \text{kg m/s}\]
The American satellite, massing \(600\) kg, had:\[p_2 = 600 \times (-7526.3) \approx -4.51578 \times 10^6 \, \text{kg m/s}\]
The total momentum before collision is calculated by:\[p_{total} = p_1 + p_2 = -2.634205 \times 10^6 \, \text{kg m/s}\]
This value helps in determining the post-collision velocity of the combined satellites.
Collision Dynamics
In collisions, understanding how momentum and velocity change helps determine outcomes, like if objects will stick together or separate.
In the scenario of the Russian and American satellites, since they collide and stick together, we call this a perfectly inelastic collision. In such conditions, objects merge, and the final momentum is the sum of their pre-collision momentums.
Using:\[v_{final} = \frac{p_{total}}{m_{total}}\]For these satellites,\[v_{final} = \frac{-2.634205 \times 10^6}{850} \approx -3099.064 \, \text{m/s}\]
The negative sign indicates the direction of the resulting velocity. After collision, their combined velocity is drastically less than the velocity required for orbit, implying they’ll lose altitude and possibly reenter Earth's atmosphere.
Satellite Motion
Satellites move in orbits due to the balance between forward velocity and downward gravitational pull. For a stable orbit, this balance must be maintained, meaning the velocity must be above a certain threshold.
When the Russian and American satellites collide, their combined velocity drops significantly below the necessary speed. This loss of speed means they can't counteract Earth's gravitational pull efficiently enough.
For the satellites to continue orbiting together, their velocity should have remained at or surpassed \(7526.3\) m/s. But with a post-collision velocity of roughly \(3099.064\) m/s, they lack this capability. Hence, they’ll move into a descending trajectory towards Earth.
Key points to remember about satellite motion include:
  • Stable orbits require maintaining a specific speed; otherwise, decay sets in.
  • Collision effects are crucial; even slight changes can alter paths significantly.
This understanding is foundational for space agencies evaluating satellite paths and avoiding collisions.

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