Calculate the speed of a satellite moving in a circular orbit about the Earth at a height of 2700×103 m. The mass of the Earth is 5.98×1024 kg and the mass of the satellite is 1400 kg.
To calculate the speed of a satellite in a circular orbit, you can use the formula:
v = sqrt((G * M) / r)
where:
v = speed of the satellite
G = gravitational constant = 6.67 x 10^-11 N(m/kg)^2
M = mass of the Earth
r = radius of the circular orbit (distance from the satellite to the center of the Earth)
First, let's convert the height of the satellite from meters to kilometers:
2700 x 10^3 m = 2700 km
Next, let's calculate the radius of the orbit by adding the height of the satellite to the radius of the Earth:
r = (radius of the Earth) + (height of the satellite)
r = 6371 km + 2700 km
r = 9071 km
Now, we can plug in the values into the formula:
v = sqrt((G * M) / r)
v = sqrt((6.67 x 10^-11 N(m/kg)^2 * 5.98 x 10^24 kg)/(9071 km * 10^3 m/km))
v = sqrt(4.02 x 10^14 N( km^2 )/(9071 x 10^3 m))
v = sqrt(4.02 x 10^14 N( km * m)/(9071 x 10^3))
v = sqrt(4.02 x 10^11 N(m))/(9071 x 10)
v = sqrt(4.02 x 10^11)/90710
v ≈ 7790 m/s
Therefore, the speed of the satellite moving in a circular orbit about the Earth at a height of 2700 x 10^3 m is approximately 7790 m/s.
To calculate the speed of a satellite in a circular orbit, you can use the formula:
v = √(G * M / r)
Where:
v = speed of the satellite
G = gravitational constant (approximately 6.674 × 10^-11 N(m/kg)^2)
M = mass of the Earth
r = distance from the center of the Earth
First, let's convert the given height of the satellite into the distance from the center of the Earth:
r = height + radius of the Earth
Given:
height = 2700 × 10^3 m (2.7 × 10^6 m)
radius of the Earth (approx.) = 6.371 × 10^6 m
Calculating the distance from the center of the Earth:
r = 2.7 × 10^6 m + 6.371 × 10^6 m
r = 9.071 × 10^6 m
Next, substitute the values into the formula:
v = √(G * M / r)
Given:
G = 6.674 × 10^-11 N(m/kg)^2
M = 5.98 × 10^24 kg
r = 9.071 × 10^6 m
Calculating the speed of the satellite:
v = √((6.674 × 10^-11 N(m/kg)^2) * (5.98 × 10^24 kg) / (9.071 × 10^6 m))
Now, solve the equation using a calculator or software:
v ≈ 7669 m/s (rounded to three significant figures)
Therefore, the speed of the satellite moving in a circular orbit about the Earth at a height of 2.7 × 10^6 m is approximately 7669 m/s.