How do the scientists calculate the correct time to launch the Mars mission so that the satellite travel time is less? How they are synchronizing the speed of satellite with respect to earth and mars ? Refer this link for more understanding: Animation of MOM and Maven launch
Well there is an extremely accurate model of the solar system and the positions of the planets and so on. So basically you have to lay Newton's equations for Mars, the probe, Earth and so on, you have an equation that models Mars path, and you have equations for your probe, so you get some x's and y's that eventually will have to match if you want to reach Mars. From there you can then calculate the values for the initial conditions of the probe -such as velocity and direction- that will satisfy you equations.
-I'm sorry my English is not very good-