A satellite orbiting Earth at an orbital radius r has a velocity v. Which represents the velocity if the satellite is moved to an orbital radius of 5r? PLEASEEE HELPPPPP
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To determine the velocity of a satellite when it is moved to a different orbital radius, we can use Kepler's Third Law of Planetary Motion.
Kepler's Third Law states that the square of the period (T) of an object orbiting a central body is proportional to the cube of the semi-major axis (r) of its orbit.
Mathematically, this can be expressed as:
T^2 ∝ r^3
Let's use this relationship to solve the problem:
1. Given that the satellite's original orbital radius is r and its velocity is v, we need to find the new velocity when the satellite is moved to an orbital radius of 5r.
2. Since we know the relationship between the period and the orbital radius, we can say that the period of the satellite when it is at an orbital radius of r is T.
3. Using this information, we can write an equation for the square of the period T^2 in terms of the original orbital radius r:
T^2 = k*r^3 (where k is a constant of proportionality)
4. Now we want to find the period when the satellite is at an orbital radius of 5r. Let's call this period T'.
5. We can write another equation for the square of the period T'^2 in terms of the new orbital radius 5r:
T'^2 = k*(5r)^3
6. We can simplify this equation:
T'^2 = k*125r^3
7. However, we know that the period T' is related to the velocity v'. We can use the equation for the period of an object in orbit to relate the original velocity v to the new velocity v':
v = (2πr) / T (original velocity)
v' = (2π(5r)) / T' (new velocity)
8. Now we can substitute the values from our equations and solve for v':
v = (2πr) / T
v' = (2π(5r)) / T'
v' = (2π(5r)) / sqrt(k*125r^3)
v' = (10πr / sqrt(125r^3)) * (1/sqrt(k))
9. Finally, we can simplify the expression:
v' = (10πr / (5sqrt(5)r)) * (1/sqrt(k))
v' = 2π / sqrt(5k)
Therefore, the velocity of the satellite when it is moved to an orbital radius of 5r is 2π / sqrt(5k).
To calculate the new velocity of a satellite when it is moved to an orbital radius of 5r, we can use the concept of specific orbital energy. The specific orbital energy (ε) of a satellite in orbit around Earth is given by the formula:
ε = -GM / (2r)
where G is the gravitational constant and M is the mass of Earth.
The specific orbital energy (ε) remains constant for an object in orbit, regardless of the orbital radius. So, the specific orbital energy at the original orbital radius, r, is the same as the specific orbital energy at the new orbital radius, 5r.
At the original orbital radius r, the velocity (v) of the satellite is given by:
v = sqrt(2GM / r)
To determine the new velocity at 5r, we need to rearrange the specific orbital energy equation to solve for v:
v = sqrt(-2GM / (2ε))
Substituting the specific orbital energy (ε) using the given formula and the original orbital radius (r), we get:
v = sqrt(-2GM / (2 * (-GM / (2r))))
Simplifying further:
v = sqrt(-2GM / (-GM / r))
v = sqrt(-2GM * (r / (-GM)))
v = sqrt(2GM * r / GM)
v = sqrt(2r)
So, the new velocity (v) of the satellite when it is moved to an orbital radius of 5r is given by sqrt(2r).