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 4r ? (1 point) 1/4 * v; 1/2 * v ○ 20 O A_{n}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 4r ? (1 point) 1/4 * v; 1/2 * v ○ 20 O A_{n}
The correct answer is 1/2 * v.
According to Kepler's third law of planetary motion, the square of the orbital period of a satellite is proportional to the cube of its orbital radius. When the orbital radius is increased to 4r, the orbital period also increases. However, the speed of the satellite remains constant at v, which means that the velocity is inversely proportional to the orbital period. Therefore, if the orbital radius is increased to 4r, the velocity of the satellite will be 1/2 * v.
If a satellite is moved to an orbital radius of 4r, the velocity can be calculated using Kepler's second law, which states that the area swept out by a satellite's position vector in a given time is constant.
According to Kepler's second law, when the satellite is moved to a larger orbital radius, its velocity will decrease. The relationship between the orbital radius and velocity is inversely proportional.
Therefore, the correct answer is 1/2 * v.
To determine the velocity of the satellite when it is moved to an orbital radius of 4r, we can use the principle of conservation of angular momentum.
The angular momentum of an object in orbit can be defined as the product of its moment of inertia (dependent on its mass distribution) and its angular velocity. In simple terms, it is a measure of how fast the object is rotating or moving in its orbit.
According to the conservation of angular momentum, the angular momentum of the satellite must remain constant when it is moved to a new orbital radius. Mathematically, this can be expressed as:
L_initial = L_final
Angular momentum is given by the formula:
L = mvr,
where m is the mass of the satellite, v is its linear velocity, and r is the orbital radius.
Since the mass of the satellite does not change and the angular momentum must remain constant, we can write:
mvr_initial = mvr_final
Canceling out the mass (m) from both sides of the equation, we get:
vr_initial = vr_final
Now, we want to find the ratio of the initial velocity (v_initial) to the final velocity (v_final) when the satellite is moved to an orbital radius of 4r. To do this, we need to understand the relationship between the initial and final radii.
The orbital velocity of a satellite can be given by the formula:
v = sqrt(GM/r),
where G is the gravitational constant, M is the mass of the Earth, and r is the orbital radius.
Let's assume that the initial orbital velocity is v_initial and the final orbital velocity is v_final.
Therefore, we have:
v_initial = sqrt(GM/r_initial)
v_final = sqrt(GM/r_final)
To find the ratio of v_initial to v_final, we divide the second equation by the first:
v_final/v_initial = sqrt(GM/r_final)/sqrt(GM/r_initial)
Using the property of square roots (sqrt(a)/sqrt(b) = sqrt(a/b)), we simplify the equation to:
v_final/v_initial = sqrt(r_initial/r_final)
Substituting the given values of r_initial = r and r_final = 4r into the equation:
v_final/v_initial = sqrt(r/4r)
v_final/v_initial = sqrt(1/4)
Taking the square root of 1/4 gives us:
v_final/v_initial = 1/2
Therefore, the velocity of the satellite when it is moved to an orbital radius of 4r is half of its initial velocity. The correct answer is 1/2 * v.