#$&*
course phy 242
june 19 around 5 pm
10) Question: `q003. Water exits a large tank through a hole in the side of a cylindrical container with vertical walls. The water stream falls to the level surface on which the tank is resting. The tank is filled with water to depth y_max. The water stream reaches the level surface at a distance x from the side of the container.Without doing any calculations, explain why there must be at least one vertical position at which the hole could be placed to maximize the distance x. Explain also why there must be distances x that could be achieved by at least two different vertical positions for the hole.
Give all the possible vertical levels of the hole.
What is the maximum possible distance x at which it is possible for a water stream to reach the level surface, and where would the hole have to be to achieve this?
YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY
YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY
Your Solution:
To allow consistency with the y_max value of 2y_1, then we must replace that value within the equation sqrt(y_1)*sqrt(y_max-y_1), which we get x= y_1.
Now the way I approached this problem is through using kinematics equation of yf= yi+v*t - .5*g*t^2 to find the time the water strikes the ground from an arbitrary point on the tank. I set the yf= 0 and find that t = sqrt(2y1/g)
Now we find the horizontal distance where the water strikes the ground using xf= xi+v*t= 0 + sqrt(2g(y2-y1))*sqrt(2y1/g)= 2 *(y2*y1-y1^2).
Now to maximize the horizontal position we take derivative of xf with respect to y1 and set it equal to zero and solve for y1, which we get it to be y1 = 0.5*y2.
@&
The horizontal range of the stream is proportional to the time of fall multiplied by the exit velocity.
Time of fall is proportional to the square root of the distance of fall. Exit velocity is proportional to the square root of the vertical distance of the water surface above the hole.
So the horizontal range is proportional to sqrt( y_1 ) * sqrt(y_max - y_1). If y_max = 2 y_1, this is not inconsistent with your solution.
If the horizontal distance was a quadratic function of hole height, then the maximum would occur halfway between. However the function is not quadratic, and it seems unlikely (but certainly possible) that the halfway height would maximize the horizontal range.
This would require a little more proof. Can you indicate your steps?
"
Self-critique (if necessary):
------------------------------------------------
Self-critique rating:
@&
Very good.
*@