Math Problem Statement

Calcola la velocità di un satellite che si muove su un’orbita circolare di raggio 6,0 × 10^8 m attorno alla Terra (M_T = 6,0 × 10^24 kg).

Solution

The image you uploaded contains a physics problem in Italian. Here's the translation:

"2. Calculate the velocity of a satellite moving in a circular orbit with a radius of 6.0×108m6.0 \times 10^8 \, \text{m} around Earth (MT=6.0×1024kg)(M_T = 6.0 \times 10^{24} \, \text{kg})."

To solve this, we can use the formula for the orbital velocity of a satellite in a circular orbit:

v=GMTrv = \sqrt{\frac{GM_T}{r}}

Where:

  • vv is the orbital velocity of the satellite.
  • GG is the gravitational constant, 6.674×1011N m2/kg26.674 \times 10^{-11} \, \text{N m}^2/\text{kg}^2.
  • MTM_T is the mass of the Earth (6.0×1024kg6.0 \times 10^{24} \, \text{kg}).
  • rr is the radius of the orbit (6.0×108m6.0 \times 10^8 \, \text{m}).

Now, substituting the values into the formula:

v=(6.674×1011)×(6.0×1024)6.0×108v = \sqrt{\frac{(6.674 \times 10^{-11}) \times (6.0 \times 10^{24})}{6.0 \times 10^8}}

Let's calculate that.

I'll do the math and get the answer. math

Given constants

G = 6.674 * 10**-11 # Gravitational constant in N m^2/kg^2 M_T = 6.0 * 1024 # Mass of Earth in kg r = 6.0 * 108 # Radius of orbit in meters

Calculating orbital velocity

v = math.sqrt((G * M_T) / r) vThe orbital velocity of the satellite is approximately 816.95m/s816.95 \, \text{m/s}.

Would you like more details about how the calculation works, or do you have any other questions?

Here are some related questions you might find interesting:

  1. What is the relationship between orbital velocity and orbital radius?
  2. How does the mass of the Earth affect the satellite's velocity?
  3. What would happen if the radius of the orbit was increased?
  4. How does the gravitational constant influence the orbital speed of satellites?
  5. Can the velocity of a satellite be used to determine its orbital period?

Tip: When calculating the orbital velocity, remember that it is derived from the balance between the gravitational force and the centripetal force required to keep the satellite in orbit.

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Math Problem Analysis

Mathematical Concepts

Orbital Mechanics
Gravitational Force
Circular Motion

Formulas

v = sqrt(G * M_T / r)

Theorems

Newton's Law of Universal Gravitation
Centripetal Force

Suitable Grade Level

Grades 11-12