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mruss
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Homework Statement
Problem Statement:
Imagine you had a very long straw, one end is in the ocean (sea
level) and the other end is in space. Air pressure at sea level
is a standard atmospheric pressure and space is a perfect vacuum.
Plot "height of water in straw" vs. time.
Variables:
h(t): height of water in straw (above sea level) at time t (in meters)
A: cross-sectional area of straw (in meters^2)
Starting conditions:
At time 0, height of water is 0: h(0) = 0
At time 0, water has no velocity or acceleration
Homework Equations
[/B]
F = ma
position(t) = position(0) + integral_0_t (velocity(0) + integral_0_t acceleration(t) dt) dt
Standard atmospheric pressure is ~101,000 kg / (m * s^2) per unit area
Density of water = 1,000.00 kg/m³
g = 9.8 m/s^2
The Attempt at a Solution
[/B]
Here is what I've tried and maybe someone can point out where I'm
going wrong.
net force = atmospheric pressure - force of gravity on mass of water in straw
Atmospheric Pressure
Standard atmospheric pressure is ~101,000 kg / (m * s^2) per unit area
Let's call A the cross-sectional area of the straw (in meters^2).
force of atmospheric pressure = 101,000 * A kg m/s^2
Force of gravity on mass of water in straw
Force of gravity on water depends on how much water is in the
straw (above sea level). Let's call h(t) the height of the
water in the straw at time t.
mass-of-water-in-straw(h(t)) = volume * density = h(t) * A * 1000 kg
force_of_gravity(h(t)) = mass-of-water-in-straw(h(t)) * g = 1000 * h(t) * A * g kg m /s^2
F(t) = 101,000 * A - 1000 * h(t) * A * g
m(t) = 1000 * A * h(t) (from mass of water equation above)
So acceleration (F / m) = (101 - g * h(t)) / h(t)
Written another way: a(t) = (101 - g * h(t)) / h(t)
Ok, great, so I think h as a function of time should be the solution to this equation:
h(t) = integral_0_t (integral_0_t a(t) dt) dt
Substitution my equation for a(t), I get:
h(t) = integral_0_t (integral_0_t (101 - g * h(t)) / h(t) dt) dt
Unfortunately, I don't know how to solve that... so I tried to
numerically approximate it. Let's just assume constant
acceleration over short periods of time dt, then we can use much
simpler equations and step through time in small dt steps.
Simpler equations:
a(t) = (101 - g * h(t-dt)) / h(t-dt)
v(t) = v(t-dt) + a(t) * dt
h(t) = h(t-dt) + v(t) * dt
with starting conditions h(0) = 0 and v(0) = 0 and a(0) = 0
When I try to calculate my initial acceleration
a(1), it's infinity (or rather, undefined) since the force of air
pressure is pushing on no mass (no water is above sea level
at the start of the problem)! There is obviously some flaw
in that logic, but I'm not sure what it is.
Any help is apprecated, thank you.
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