Question

Remember to show all work in order to receive full credit. Miracle answers or equations with just numbers will not receive full credit. 1. Ignoring air resistance, water exiting a fire hose satisfies the expression where g is a positive constant (magnitude of the acceleration due to gravity) m is the slope of the fire hose at the ground v is a positive constant (initial speed of the water stream out of the fire hose) x is the horizontal position of the water e . y is the vertical position of the water The initial position of the water is at the origin of the coordinates, (0,0) All answers should be in terms of parameters of the problem m, g, d, v and numbers, and please show your algebra/calculus! ® (A) For a given value of m, determine the distance (x) from the nozzle at which the water eaches the ground. (B) Determine the value of m for which the water reaches the ground at the greatest distance from the nozzle of the hose. HINT: In calculus, finding the "maximum" or "minimum" usually means something specific. Here, consider what is that is being maximized and what it is that is varied... this will give what needs to be calculated! (C) For a given value of m, determine the value of x for which the height of the water (y) is maximum (D) Find the value of m for which the water reaches the greatest height on a vertical wall a given distance d from the nozzle re ose

Answer 1

Given Data Equation of water existing from the hose, y = mx - g/2 (1 + m^2) (x/v)^2. (A) The water reaches the ground the y coordinate is Zero . Substitute 0 for y in the equation of water . 0 = mx - g/2 (1 + m^2) (x/v)^2 x{m - g/2 (1 + m^2) (x/v^2)} = 0 solve the above expression to obtain the roots. x = 0 or m = g/2 (1 + m^2)(x/v^2) x = 2mv^2/g(1 + m^2) Since the initial Position of the water is origin, so the distance from the nozzle at which the water reaches the ground is, d_max = 2_mv^2/g(1 + m^2) Here, the distance reached is d_max. Thus for a given value of m the distance from the nozzle at which the water reaches the ground is 2_mv^2/g(1 + m^2) (B) Write the expression for the maximum distance reached. d_max = 2_mv^2/g(1 + m^2) Since m is slope of the hose, so write the slope in angular form. m = tan theta