Question

Два протони з кількома енергіями T=20 ГеВ кожний рухаються так, що найменша і найбільша відстані від центра Землі відповідно r1=6600 км і r2=6700 км. Обчислити момент імпульсу L супутника відносно центра Землі. Якщо можна, із покроковим поясненням, будь ласка.

Answer 1

Calculation of Satellite's Angular Momentum

To calculate the angular momentum (L) of a satellite relative to the center of the Earth, we can use the formula:

L = mvr

Where: - L is the angular momentum - m is the mass of the satellite - v is the velocity of the satellite - r is the distance between the satellite and the center of the Earth

In this case, we are given that the satellite consists of two protons, each with an energy of T = 20 GeV. We need to find the angular momentum of the satellite.

To calculate the angular momentum, we need to determine the velocity of the satellite. We can use the relativistic energy-momentum relation to find the velocity.

The relativistic energy-momentum relation is given by:

E^2 = (mc^2)^2 + (pc)^2

Where: - E is the total energy of the particle - m is the rest mass of the particle - c is the speed of light in a vacuum - p is the momentum of the particle

In this case, the energy of each proton is T = 20 GeV. Since the rest mass of a proton is approximately 938 MeV/c^2, we can calculate the momentum of each proton using the relativistic energy-momentum relation.

Let's calculate the momentum of each proton:

E^2 = (mc^2)^2 + (pc)^2 (20 GeV)^2 = (938 MeV/c^2)^2 + (pc)^2 pc = sqrt((20 GeV)^2 - (938 MeV/c^2)^2)

Using the given values, we can calculate the momentum of each proton:

pc = sqrt((20 GeV)^2 - (938 MeV/c^2)^2) = 19.999 GeV

Now that we have the momentum of each proton, we can calculate the velocity of the satellite. The momentum of a particle is given by:

p = mv

Where: - p is the momentum - m is the mass of the particle - v is the velocity of the particle

In this case, the mass of each proton is approximately 1.67 x 10^-27 kg. We can calculate the velocity of each proton using the momentum:

p = mv 19.999 GeV = (1.67 x 10^-27 kg)v

Solving for v:

v = (19.999 GeV) / (1.67 x 10^-27 kg) = 1.198 x 10^19 m/s

Now that we have the velocity of the satellite, we can calculate the angular momentum using the formula:

L = mvr

In this case, the mass of the satellite is the sum of the masses of the two protons:

m = 2 * (1.67 x 10^-27 kg) = 3.34 x 10^-27 kg

Using the given values, we can calculate the angular momentum:

L = (3.34 x 10^-27 kg)(1.198 x 10^19 m/s)(r)

We are also given the minimum distance from the center of the Earth (r1 = 6600 km) and the maximum distance from the center of the Earth (r2 = 6700 km). To calculate the angular momentum, we need to use the average distance between these two values:

r = (r1 + r2) / 2

Substituting the values, we can calculate the angular momentum:

L = (3.34 x 10^-27 kg)(1.198 x 10^19 m/s)((6600 km + 6700 km) / 2)

Converting the average distance to meters:

L = (3.34 x 10^-27 kg)(1.198 x 10^19 m/s)((6600 km + 6700 km) / 2)(1000 m/km)

Calculating the angular momentum:

L = 4.021 x 10^-8 kg m^2/s

Therefore, the angular momentum of the satellite relative to the center of the Earth is 4.021 x 10^-8 kg m^2/s.

Please note that the calculations provided are based on the given information and assumptions.