Thuswehaveamodifiedformulationwhichgives 32

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Unformatted text preview: indrop To illustrate the process, we consider the problem of a raindrop falling from a cloud at moderate height. Goal – To find time the taken by a raindrop falling from a cloud at moderate height to reach the ground. Real world consists of rain drop, cloud, ground, surrounding environment. Observations: (i) The velocity of raindrop increases as the distance travelled increases. (ii) A large raindrop takes about 40 sec. to reach ground from a cloud at the height of 1024 sec. (experimental observation) Simplifications/ Idealization Raindrop is a particle falling from rest along a straight line. The variables here are: time, distance, velocity of the raindrop. Mathematical Formulation: If x(t) is the distance travelled in time t by the rain drop after its fall from cloud, then its velocity is rate of change of x with time, i.e., . Now from the assumption we have with x = 0 at t = 0 (here k is the constant of proportionality). Solving this we get the x = 0 for all time ‐ the raindrop is not moving at all. This is not correct. So we have to correct the things. It may be pointed out that while observation is correct, the drop is falling under gravity and Galileo had observed that ‘An object falling from moderate height under gravity gains an extra 32 ft/ sec in velocity in each second.’ Thus we have a modified formulation, which gives: 32 As initial velocity 0 (rain drop falls from rest), we have 32 , 0 0. This gives time to cover a distance of 1024 ft is 8 sec. – th...
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