A planet orbits the sun at a distance of 2.87 x 10^9 km. If the mass of the sun is 1.99 x 10^30 kg, find the orbital period of the planet. Then, calculate the orbital speed of the planet.
I don't understand
R = 2.87 * 10^12 meters
G =6.67*10^-11
G m M = m v^2/R
so
v^2 = G M R
find speed v
then 2 pi R/v = period
To find the orbital period of the planet, you can use Kepler's Third Law, which states that the square of the orbital period of a planet is proportional to the cube of the average distance (radius) between the planet and the sun.
The formula for Kepler's Third Law is:
T^2 = (4π^2 * r^3) / (G * M)
where:
T is the orbital period of the planet,
r is the average distance (radius) between the planet and the sun,
G is the gravitational constant (approximately 6.674 x 10^-11 N(m/kg)^2),
M is the mass of the sun.
Using the given values:
r = 2.87 x 10^9 km = 2.87 x 10^12 meters
M = 1.99 x 10^30 kg
G = 6.674 x 10^-11 N(m/kg)^2
Plugging these values into the equation, we have:
T^2 = (4π^2 * (2.87 x 10^12)^3) / (6.674 x 10^-11 * 1.99 x 10^30)
Simplifying:
T^2 = (4π^2 * 2.0819233 x 10^37) / (1.3284426 x 10^19)
Taking the square root to isolate T:
T = √[(4π^2 * 2.0819233 x 10^37) / (1.3284426 x 10^19)]
Calculating this expression will give you the orbital period of the planet.
Next, to calculate the orbital speed of the planet, you can use the formula for orbital speed:
v = (2π * r) / T
where:
v is the orbital speed of the planet,
r is the average distance (radius) between the planet and the sun,
T is the orbital period of the planet.
Plugging in the values:
r = 2.87 x 10^9 km = 2.87 x 10^12 meters
T is the orbital period that you calculated earlier.
Using these values in the equation will give you the orbital speed of the planet.
Google Newton, universal gravitation F = G m M /R^2
and gravitational force = mass times centripetal acceleration (m v^2/R)
sorry, made algebra error
R = 2.87 * 10^12 meters
G =6.67*10^-11
G m M/R^2 = m v^2/R
so
v^2 = G M / R
find speed v
then 2 pi R/v = period