3.59) A water hose is used to fill a large cylindrical tank of diameter D and he

Question

3.59) A water hose is used to fill a large cylindrical tank of diameter D and height 2D. The hose shoots at 45 degrees above the horizontal from the same level as the base of the tank and a distance 6D from the tank. For what range of launch speeds will the water enter the tank? Give your answer in terms of D and g.

Explanation / Answer

Since this is a projectile motion problem we have to use the kinematic equations for the motion along the x-axis and y-axis. For motion along the x-axis, ?x = v0cos(45) t ? t = ?x/(v0cos(45)) >We solve for t so that we can plug it into the y-axis kinematic equation. For motion along the y-axis, ?y = v0sin(45)t - ½ gt^(2) = v0sin(45)[?x/(v0cos(45))] - ½ g[?x/(v0cos(45))]^2 = ?xtan(45) - g?x^(2)/[2v0^(2)(cos(45))^(2)] >What you are trying to get ride of is t so the equation is expressed in ?x, g and v0 since the question is asking the minimum and maximum speed and it gives us the distance, ?x, and g is constant. From ?y solve for v0 and plug in 45 degrees to get, v0 = ?x*sqrt[g/(?x - ?y)] Then with ?x = 6D and ?y = 2D, v0min = 3*sqrt[gD] Then with ?x = 7D and ?y = 2D, v0max = 7*sqrt[gD/5]

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