"MODELLING RAINFALLRationale: Looking for a better perspective for my mathematical exploration I foundweather to be a different idea. It was difficult to think regarding the possibilityto predict future weather with the help of mathematics. Hence, after doing lot ofresearch and with teachers help I could decide to do my exploration on maths asevery year India faces lots of difficulty due to either heavy rainfall or lessrainfall. But gain question arises, how it is possible to model rain and how tomove my research further so that I could structure my exploration in a correctway. Serious consequences for the households, industries as well as environment areformed due to high intensity rainfall. It results in flood which causes damage tothe urban scale. The estimation of the models is still a subject to think. Therefore, I have decided to do my exploration on rainfall modelling. Thisexploration helps us to find the mathematical significance in rainfall modelling.Hence, it will be quite possible to predict as well as estimate future rainfall bywhich some precautions can be taken to overcome the situation.Research Question:Can we predict future rainfall with Modelling Rainfall?Aim:My exploration here is to investigate Modelling Rainfall.Introduction:The forecasting of weather is a scientific as well as technological activity whichhelps us in our day to day life. We mainly experience four seasons in a year butrainfall not comes in its season but we can see them in any of the season andthus we are familiar with the phenomenon on a regular basis. Therefore, itbecomes very important for us to know about it. The models provide usinformation of the future which we apply in different ways according ourrequirement. Hence, this exploration helps us to learn regarding model for rain.First of all we need to know the factors related to rainfall so that we cancorrectly model. The measurement of rainfall is done on the basis of its amountthat is its height (millimetre per hour) is measured because its intensity is always different. Intensity can be one factor as there are many factors forrainfall.First Model (not accounting for air resistance): The gravitational force can be considered as one of the factor as any objectwhen left from above it falls towards the ground. Therefore, I studied aboutNewton’s second law of motion which is regarding the falling objects. Hereforce is equal to the mass times acceleration.This can be written as F = maHere, F is force; m is mass of the object we use as well as its acceleration. We can use this Newton’s Law of Motion to model rainfall along with itsacceleration. It is considered that acceleration, a, is velocities first derivative, v,whereas, s is the second derivative of position. Hence, we can write this as; 2 ? ? ? ? ? = =2 ? ? ? ? For any object that falls we can write as; ? = ?Thus we get, ? ? = ? = ?? ?We assume here raindrop starting from 500 meters in the air as it is difficult tofind out the accurate height. Hence, for raindrop we get differential equationhaving constant acceleration;? ? = ?? ? ? ? = ? ? ? ? ? = ? ? ?? = ? ? + ?For constant of integration, explanation given above is used here as; ? 0 = 0,? = 00 = ? 0 + ?? = 0Hence,? = ? ? Therefore, now another equation can also be used to move further such as;? ? = ?? ? Now the basic integration is used here for solving the differential equation as;? = ? ?? ? ? =? ? ? ? = ? ?? ? ? ? = ? ? ? ? ? ? = ? ? ? ?? 2 ? = ? + ?2 Therefore, for determining the constant of integration, we use the earlierconditions again. Here we assume 0 for solving the equation. Hence, ? 0 = 0, ? 0 = 0Therefore, ? 20 = 0 + ?2 ? = 0So we get;? 2 ? = ?2 -2 As per our assumption, ? =500m and ? = 9.81? ? . These can be substitutedin the equation given above for solving ? ;9.81 2 500 = ?2 2 1000 = 9.811?1000 2 ? =9.81 1000 ?9.81 = 10.096Hence, for the determination of the raindrops velocity falling on the ground,first equation can be used as; ? = ? ?? = 9.81 * 10.096 ? = 99.044Therefore, our assumption regarding the distance from the rain falls was 500-2 metres according to which we experience constant acceleration of 9.81? ? .As per these figures, time taken by raindrop to fall on the ground is 10.096seconds and would hit the ground at around 99.044 meters per second. Here we find velocity big. This model can be taken just for our understanding aswe have just assumed rain drop to fall from 500 metres. It was just thegravitational force that was taken whereas in reality it is the air resistance whichneeds to be considered. Again for the constant acceleration model it can be considered that it can workfor any size of the raindrop as acceleration as well as velocity none of them aredependent on mass. If we see accelerations equation that is;? ? = ?? ? We get to know that determination of the terminal velocity of the raindrop is not-2 possible, because ? = 9.81? ? as well as acceleration both can never be zero.Hence, raindrop will never stop accelerating which is not actually possible. Second Model (Small raindrops): As per the research a very small drop having diameter of ? = 0.008? ?experiences a drag force that is proportional to velocity. Hence, given is theforce proportional to acceleration we get;? ? ? ? ?? ? ? ? ? ? ? ? Hence,? = ? ?? ? ? ? -1 This shows that, ? = 12. 2?-2 Acceleration as given above is constant for air resistance, ? = 9.81? ? .Therefore, we need to find a new differential equation for the model.Acceleration is hence given as;? ? ? =? ? ? ? = ? - ? ?? ? Raindrops terminal velocity can be found by using the above equation. Hence,we get terminal velocity from the model;0 = ? - ? ?? = ? ? "