Steel can of H height and A area, full of water.
Hole in the bottom of the can of \( A_2 \) area.
Find time it takes to drain the can.
Yadda yadda Bernouli's Eq applies.
V = velocity, P = pressure, z = height, Row = density.
\(
P_1 + 1/2RowV^2 + Row*g*z = P_2 + 1/2RowV_2^2 + Row*g*z_2
\)
Pressures are equal outside the can, Row cancels, V_1 is negligible and \( z_2 \) = 0.
\(
gz=1/2V_2^2
V_2 = sqrt{2gz}
\)
Relationship between volume out of the can and change in height, feel like I'm wrong here:
\(
A_1 H/t=A_2 V_2
H = A_2 /A_1 *V_2 t
dh/dt = A_2 /A_2 V_2
\)
Regardless of whether that is right, I think this is my differential equation:
\(
dh/dt = -A_2 /A_1 V_2
dh/dt = -A_2 /A_1 sqrt{2gh}
\)
Separate Variables
\(
dh/sqrt{h}=-A_2 /A_1 sqrt{2g}dt
\int 1/sqrt{h} dh = \int -A_2 /A_1 sqrt{2g}dt
2sqrt{h} = -A_2/A_1 sqrt{2g} t +C
h = ((-A_2/A_1 sqrt{2g} t)/2 +C)^2
\)
Solve for t when h = 0.
\(
0 = ((-A_2/A_1 sqrt{2g} t)/2 +C)^2
2C*A_1/A_2 * 1/sqrt{2g} = t
\)
Doesn't feel right, and my calculator is graphing it with an absurdly long time until empty.
Hole in the bottom of the can of \( A_2 \) area.
Find time it takes to drain the can.
Yadda yadda Bernouli's Eq applies.
V = velocity, P = pressure, z = height, Row = density.
\(
P_1 + 1/2RowV^2 + Row*g*z = P_2 + 1/2RowV_2^2 + Row*g*z_2
\)
Pressures are equal outside the can, Row cancels, V_1 is negligible and \( z_2 \) = 0.
\(
gz=1/2V_2^2
V_2 = sqrt{2gz}
\)
Relationship between volume out of the can and change in height, feel like I'm wrong here:
\(
A_1 H/t=A_2 V_2
H = A_2 /A_1 *V_2 t
dh/dt = A_2 /A_2 V_2
\)
Regardless of whether that is right, I think this is my differential equation:
\(
dh/dt = -A_2 /A_1 V_2
dh/dt = -A_2 /A_1 sqrt{2gh}
\)
Separate Variables
\(
dh/sqrt{h}=-A_2 /A_1 sqrt{2g}dt
\int 1/sqrt{h} dh = \int -A_2 /A_1 sqrt{2g}dt
2sqrt{h} = -A_2/A_1 sqrt{2g} t +C
h = ((-A_2/A_1 sqrt{2g} t)/2 +C)^2
\)
Solve for t when h = 0.
\(
0 = ((-A_2/A_1 sqrt{2g} t)/2 +C)^2
2C*A_1/A_2 * 1/sqrt{2g} = t
\)
Doesn't feel right, and my calculator is graphing it with an absurdly long time until empty.