A water tank of height hhas a small hole at height y.The water is replenished to

Question

A water tank of height hhas a small hole at height y.The water is replenished to keep hfrom changing. The water squirting from the hole has rangex.The range approaches zero as y o 0 because the water squirts right onto the table. The rangealso approaches zero as y o h because the horizontal velocity becomes zero. Thus theremust be some height ybetween 0 and hfor which the range is a maximum. a. Find an algebraic expression for the flow speed v with which the water exits the hole at heighty.? b. Find an algebraic expression for the range of a particleshot horizontally from height ywith speed v.? c. Combine your expressions from parts A andB. Find the maximum range x_{{ .?

Explanation / Answer

let teh horizontal velocity of the water stream be Vx = sqrt(2gy) the time taken to reach the ground = h - y =0.5 * g *t^2 t = sqrt (2 * (h-y)/g) the range of the stream would be = Vx * t = 2 * sqrt ((h-y)*y) for finding the maximum range we differentiate the equation wrt y d(hy-y^2)/dy = 0 solving this we get , y = 0.5 * h thus the maximum range = 2 * sqrt (h - 0.5 * h) * 0.5 * ) maximum range = h

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