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.

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.

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: \(p_2 = (600\, \text{kg})(-7526.3\, \text{m/s}) = -4.51578 \times 10^6\, \text{kg m/s}\)

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 \(m_{total}= 250\,\text{kg} + 600\,\text{kg}= 850\,\text{kg}\). \(v_{final} = \frac{-2.634205 \times 10^6\,\text{kg m/s}}{850\,\text{kg}} = -3099.064\,\text{m/s}\)

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.