Suppose one of the Global Positioning System satellites has a speed of 4.50 km/s at perigee and a speed of 3.52 km/s at apogee. If the distance from the center of the Earth to the satellite at perigee is 2.21×104 km, what is the corresponding distance at apogee?
Vp/Va = r(a)/r(p)
To solve this problem, we can use the fact that the angular momentum of the satellite remains constant throughout its orbit. The angular momentum is given by the equation:
L = mvr
where L is the angular momentum, m is the mass of the satellite, v is the velocity of the satellite, and r is the distance from the center of the Earth.
Since the satellite is the same throughout its orbit, we can equate the angular momentum at perigee (L_perigee) to the angular momentum at apogee (L_apogee):
L_perigee = L_apogee
Now, let's denote the mass of the satellite as m, the velocity at perigee as v_perigee, the velocity at apogee as v_apogee, the distance from the center of the Earth at perigee as r_perigee, and the distance from the center of the Earth at apogee as r_apogee.
Using the equation for angular momentum, we have:
m * v_perigee * r_perigee = m * v_apogee * r_apogee
We are given that the velocity at perigee, v_perigee, is 4.50 km/s and the distance from the center of the Earth at perigee, r_perigee, is 2.21×10^4 km. We need to find the corresponding distance at apogee, r_apogee.
To find r_apogee, we rearrange the equation:
r_apogee = (v_perigee * r_perigee) / v_apogee
Substituting the given values, we have:
r_apogee = (4.50 km/s * 2.21×10^4 km) / 3.52 km/s
Calculating this expression, we can find the corresponding distance at apogee.
To solve this problem, we need to apply Kepler's second law, which states that the line joining a planet and the sun sweeps out equal areas in equal intervals of time. This means that the satellite will move faster when closer to the Earth (perigee) and slower when farther away (apogee).
Let's consider the perigee position first:
Given:
Velocity at perigee (Vp) = 4.50 km/s
Distance from the center of the Earth at perigee (Rp) = 2.21 × 10^4 km
We can calculate the angular momentum (L) of the satellite at perigee using the formula:
L = m * v * r
Where:
m = mass of the satellite (which we can neglect, as it cancels out in the subsequent calculations)
v = velocity at perigee = 4.50 km/s
r = distance from the center of the Earth at perigee = 2.21 × 10^4 km
Now let's consider the apogee position:
Velocity at apogee (Va) = 3.52 km/s (given)
Distance from the center of the Earth at apogee (Ra) = ?
Since the angular momentum (L) remains constant, we can equate the angular momentum at perigee (Lp) to the angular momentum at apogee (La):
Lp = La
Therefore, we have:
m * Vp * Rp = m * Va * Ra
The mass of the satellite (m) cancels out, and we can solve for Ra:
Ra = (Vp * Rp) / Va
Now, substitute the given values:
Ra = (4.50 km/s) * (2.21 × 10^4 km) / (3.52 km/s)
Simplify this equation to find the corresponding distance at apogee (Ra).