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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 =**0

**i +**g

**(-j) +**0

**k;**

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*cosd(90-θ); g*t + Vb*sind(90-θ); 0**

**Vb(t)dt = Vb(t) = [-****]dt;**

=>

**R(t) =**(Vb*cosd(90-θ)*t + c4)

*****(-g/2*t^2 + Vb*sind(90-θ)*t + c5)

**(i)**+*****c6

**(j)**+*****impose initial conditions:

**(k)**

**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 )

*****(-g/2*t^2 + Vb*sind(90-θ)*t )

**(i)**+*****0

**(j)**+*****

**(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:

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)

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.