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ltkach2015
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Homework Statement
Water is sprayed at an angle of 90° from the slope at 20m/s. Determine the range R.
PLEASE SEE ATTACHMENT
Homework Equations
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Kinematic Equations:
acceleration: A = 0i +g(-j) + 0k;
velocity: dV/dt = A;
position: dR/dt = V;
Origin set at the point spout of water.
Angle of the plane: θ = atand(3/4) = 36.869°
Magnitude of acceleration: g = 9.81;
Initial Conditions
Velocity: Vb(t=0) = 10; => Vb(t=0) = Vb*cosd(90-θ)*(-i) + Vb*sind(90-θ*(-j) + 0*(-k)
Position: Rb(t=0) = 0*(i) + 0*(j) + 0*(k)
The Attempt at a Solution
1) integrate acceleration: A = [0; -g ;0] = dVb/dt∫dVb = ∫ [0; -g; 0] dt
=> Vb(t) = c1*(i) + (-g*t + c2)*(-j) + c3*(k);
impose initial conditions: Vb(t=0) = Vb*cosd(90-θ)*(-i) + Vb*sind(90-θ)*(-j) + 0*(-k)
=> c1 = Vb*cosd(90-θ); c2 = Vb*sind(90-θ); c3 = 0;
2) integrate velocity:
∫dR = ∫Vb(t)dt = Vb(t) = [-Vb*cosd(90-θ); g*t + Vb*sind(90-θ); 0]dt;
=> R(t) = (Vb*cosd(90-θ)*t + c4)*(i) + (-g/2*t^2 + Vb*sind(90-θ)*t + c5)*(j) + c6*(k)
impose initial conditions: Rb(t=0) = 0*(i) + 0*(j) + 0*(k)
=> c4 = 0; c5 = 0; c6 = 0;
3) List of Derived Equations:
R(t) = (-Vb*cosd(90-θ)*t )*(i) + (-g/2*t^2 + Vb*sind(90-θ)*t )*(j) + 0*(k)
Vb(t) = (-Vb*cosd(90-θ))*(i) + (-g*t + Vb*sind(90-θ))*(j) + 0*(k)
4) Equation of Inclined Plane:
f(x) = 3/4*x
5) Parametrize the position function:
x(t) = (-Vb*cosd(90-θ)*t )
=> t = x(t)/(Vb*cosd(90-θ)
y(x(t)) = Vbsind(90-θ)*{x(t)/(Vb*cosd(90-θ))} - g/2*({x(t)/(Vb*cosd(90-θ))}^2
6) Find the Intersection of the two plane curves.
7) Use magnitude of x and y to find Range: (probably didn't say that correctly)
Range = sqrt( xi^2 + yi^2)I would appreciate all the help I can get. Thank you.