mars' orbit has a diameter 1.52 times that of the earth's orbit. how long does it take mars to orbit the sun?(period of the earth =365 days)
We know that an object in orbit around another has centrifugal force F = (mv^2)/r
We know that the force on an object due to gravity is F=GMm/r^2 where G is the universal gravitational constant, M is the mass of the sun and m is the mass of the orbiting object (in this case Mars).
Equating these two gives us
GMm/r^2 = (mv^2)/r
Rearanging for v
Now, velocity = distance / time.
So for an object moving in a circle of diameter D and taking time t to do it,
v=pi*D/t. note D=2r, so v=2pi*r/t.
Rearange for t:
t = 2*pi*r/v.
Sub in v from our original equation and we have
If Mars' orbit is 1.52* Earth orbit, then plug in the numbers
(G = 6.67*10^(-11) Earth Orbit = 1.5*10^11
M = 2*10^30 Mars Orbit r = 1.52*Earth Orbit
This comes out as about 685days.
You want an orbit with perihelion at Earth's orbit (q = 1.0 AU) and an aphelion at Mars' orbit (let's say Q = 1.5 AU, you might want to use the actual semi-major axis of Mars' orbit). Now 2a = q + Q so for our transfer orbit 2a = 1.0 + 1.5 therefore a (for the transfer orbit) = 1.25. From that point you should be away because you can use this 'a' in the vis-viva equation and in Kepler's Third Law T^2 = a^3
Average orbital speed of Mars = 24.077 km/s
Average orbital speed of Earth = 29.783 km/s
Earth's orbit = Earth's velocity x Time taken for Earth to orbit the sun =365 x 24 x 60 x 60 x 29.783 =939236688 km
Time taken for Mars to orbit the sun = Mars's orbit / Mars's average orbital speed = 1,52 x Earth's orbit / Mars's average orbital speed = 59294752 s = 686,282 Earth days
668 days from memory