Problems on Mental Math
Q1) Assume that the Earth is made up only from neutrons. Since the mass of the Earth is ~ 1024 kg, what would be the number of neutrons required to make up the earth, if the mass of a neutron is ~ 10-27 kg?
Q2) Assume that you are to design a rocket to travel from the Earth to Venus (distance ~ 0.3 AU). The engine of the rocket has an energy efficiency of 30%, i.e. only 30% of the fuel supplied is converted into energy, while the rest is wasted. Also, you may assume that the rocket has a uniform velocity, i.e. it burns the same amount of fuel to cover the same distance, say 1 kg per 100 km. Calculate the amount of fuel required to reach Venus.
Q3) If you want the same rocket as in Q2) to travel to Mercury from the Earth (distance ~ 0.6 AU). It is apparent that either the rocket will have to carry more fuel, or since it is beyond its capacity, it will have to carry fuel of greater efficiency. As we saw the fuel used in Q2) has 30% efficiency, what should be the efficiency of the fuel to carry our rocket to Mercury?
Q4) Ants are usually 5 mm across. Hence their antists (I mean ant scientists) define the unit of distance as antfoot. The world convention between ant and human scientists settled the unit such that 1 antfoot = 2 mm. Now if an ant travels the length of India from Kashmir to Kanyakumari, then she would travel say ‘N’ antfeet. Find a distance in the solar system, which when measured in human Feet is comparable (order of magnitude) to ‘N’ human feet.
Q5) If a spacecraft travels at 10,000 km/s when in deep space (i.e. far away from the gravity of the planets in the solar system) and is proceeding towards the alpha centauri system, then how long will it take to reach the