1- If a satellite orbiting the Earth at a height of 150 km (93 miles) above the surface of the earth.
Calculate the speed, acceleration and orbital period of the satellite. Using these given constants:
M(earth) = 5.98 x 10^24 kg, R(earth) = 6.37 x 10^6 m).
F = G m M/r^2
G = 6.67*10^-11
m is mass of our thing
M = 5.98*10^24
r = 6.37*10^6 + 150,000 = 6.37*10^6 + 0.150 *10^6 = 6.52*10^6
so
F = m * 6.67*10^-11 * 5.98*10^24 / 42.5*10^12
= m * 9.38
{{ note - if we were at earth surface it would be about m * 9.81 , we did not go up much :) }}
Now the problem:
F = mass * centripetal a = m * v^2/r
so
9.38 = v^2/r = v^2 / 6.52*10^6
v^2 = 9.38 * 6.52 * 10^6 meters/second
solve for v
then a = v^2/r
T = 2 pi r / v
To calculate the speed, acceleration, and orbital period of the satellite, we can use the following formulas:
1. Speed of Satellite:
The speed of the satellite can be calculated using the formula:
v = √(G * M / r)
where,
v = speed of satellite,
G = gravitational constant (6.67430 x 10^-11 m^3/kg/s^2),
M = mass of the Earth (5.98 x 10^24 kg),
r = radius of the satellite's orbit (R(earth) + height of satellite).
Plugging in the values, we get:
r = distance from the Earth's center to the satellite = R(earth) + height of satellite
= (6.37 x 10^6 m) + (150 x 10^3 m) = 6.52 x 10^6 m
Now, we can calculate the speed of the satellite:
v = √(G * M / r)
= √((6.67430 x 10^-11 m^3/kg/s^2) * (5.98 x 10^24 kg) / (6.52 x 10^6 m))
Using a calculator, we find:
v ≈ 7637 m/s (rounded to 3 significant figures)
2. Acceleration of Satellite:
The acceleration of the satellite can be calculated using the centripetal acceleration formula:
a = (v^2) / r
where,
a = acceleration of the satellite.
Plugging in the values, we get:
a = (v^2) / r
= (7637 m/s)^2 / (6.52 x 10^6 m)
≈ 8.94 m/s^2 (rounded to 3 significant figures)
3. Orbital Period of Satellite:
The orbital period of the satellite can be calculated using the formula:
T = 2π * √(r^3 / (G * M))
where,
T = orbital period of the satellite.
Plugging in the values, we get:
T = 2π * √(r^3 / (G * M))
= 2π * √((6.52 x 10^6 m)^3 / ((6.67430 x 10^-11 m^3/kg/s^2) * (5.98 x 10^24 kg)))
Using a calculator, we find:
T ≈ 5584 seconds (rounded to the nearest second)
Therefore, the speed of the satellite is approximately 7637 m/s, the acceleration is approximately 8.94 m/s^2, and the orbital period is approximately 5584 seconds.
To calculate the speed, acceleration, and orbital period of a satellite orbiting the Earth, we can use the following formulas:
1. Speed of the satellite:
The speed of the satellite can be calculated using the formula for circular motion: v = √(G * M / r), where v is the speed, G is the gravitational constant, M is the mass of the Earth, and r is the radius of the orbit.
2. Acceleration of the satellite:
The acceleration of the satellite can be calculated using the formula: a = v^2 / r, where a is the acceleration, v is the speed, and r is the radius of the orbit.
3. Orbital period of the satellite:
The orbital period of the satellite can be calculated using the formula: T = 2π * (r / v), where T is the orbital period, π is a constant (approximately 3.14159), r is the radius of the orbit, and v is the speed.
Using the given values:
Height above the surface of the Earth, h = 150 km = 150,000 m
Radius of the Earth, R(earth) = 6.37 x 10^6 m
Mass of the Earth, M(earth) = 5.98 x 10^24 kg
Gravitational constant, G = 6.67430 x 10^-11 N(m/kg)^2
Calculating the radius of the satellite's orbit:
The radius of the satellite's orbit, r = R(earth) + h
Substituting the values:
r = 6.37 x 10^6 m + 150,000 m = 6.52 x 10^6 m
Calculating the speed of the satellite:
v = √(G * M / r)
v = √((6.67430 x 10^-11 N(m/kg)^2) * (5.98 x 10^24 kg) / (6.52 x 10^6 m))
v ≈ 7630 m/s
Calculating the acceleration of the satellite:
a = v^2 / r
a = (7630 m/s)^2 / (6.52 x 10^6 m)
a ≈ 1.15 m/s^2
Calculating the orbital period of the satellite:
T = 2π * (r / v)
T = 2π * (6.52 x 10^6 m / 7630 m/s)
T ≈ 5479 seconds or 1.52 hours
Therefore, the speed of the satellite is approximately 7630 m/s, the acceleration is approximately 1.15 m/s^2, and the orbital period is approximately 5479 seconds or 1.52 hours.