How does the force due to gravity on mars compare to the force due to gravity on earth? , and using newton's second law, explain how a rocket can have an increase in acceleration over time.

As accurately described by Einstein's theory of relativity, gravity is not necessarily a force, but a consequence of the curvature of space time that is caused by the uneven distribution of mass. But this could also be approximated by Newton's Law of Universal Motion. Gravity is a force acting upon two objects with masses at a certain amount of distance. So, the greater the mass, and the closer the objects are, the greater is the force of gravity. So, if you compare the gravity on Mars compared to Earth at a given distance, compare their masses. The mass of Earth is 5.972 × 10^24 kg while that of Mars is 6.39 × 10^23 kg. So, the gravity on Earth is much greater because Earth is heavier.

For the second question, let's base on Newton's Second Law of Motion which states: the force exerted on or by the object, is equal to its mass times acceleration. In equation, F = ma. So, if you want to increase acceleration, the force should be greater, while the mass should be lighter.