Two small spaceships, each with mass m = 2000 kg, are in the circular Earth orbit of the figure, at an altitude h of 420 km. Igor, the commander of one of the ships, arrives at any fixed point in the orbit 90.0 s ahead of Picard, the commander of the other ship. At point P , Picard fires an instantaneous burst in the forward direction, reducing his ship's speed by 1.30%. After this burst, he follows the elliptical orbit shown dashed in the figure. How much earlier than Igor will Picard return to P?
Point p is located at the height above earth at 180 degrees.
To determine how much earlier Picard will return to point P compared to Igor, we need to calculate the time it takes for each spaceship to complete one orbit.
Let's start by finding the radius of the circular Earth orbit at altitude h = 420 km. We know that the radius of the Earth is approximately 6,371 km. Since the altitude is given in kilometers, we need to convert it to meters by multiplying by 1000:
Altitude h = 420 km = 420,000 meters
The radius of the Earth orbit is then:
R = radius of Earth + altitude = 6,371 km + 420 km = 6,791 km = 6,791,000 meters
Next, we can calculate the orbital speed of the spaceships in the circular orbit. The orbital speed can be found using the formula:
v = sqrt(G * M / R)
Where:
- G is the gravitational constant (approximately 6.67 × 10^(-11) N(m^2/kg^2))
- M is the mass of the Earth (approximately 5.97 × 10^24 kg)
- R is the radius of the circular orbit
Plugging in the values, we have:
v = sqrt((6.67 × 10^(-11) N(m^2/kg^2)) * (5.97 × 10^24 kg) / (6,791,000 meters))
Calculating this gives us the orbital speed of the spaceships.
Now, let's calculate the time it takes for Igor to complete one orbit. Since Igor arrives at any fixed point in the orbit 90.0 s ahead of Picard, we can conclude that Igor's orbital period is 90.0 s shorter than Picard's.
The orbital period T can be calculated using the formula:
T = (2 * pi * R) / v
Where:
- T is the orbital period
- pi is a mathematical constant approximately equal to 3.14159
- R is the radius of the circular orbit
- v is the orbital speed
Calculating this gives us the orbital period for Picard.
Finally, to determine how much earlier Picard will return to point P compared to Igor, we subtract Igor's orbital period from Picard's orbital period:
Time difference = Picard's orbital period - Igor's orbital period
This will give us the desired result.