How much the gas has increased the speed of the plug?

[kaspis245]

Homework Statement

The cannon is made from V=30ml test tube completely closed with a plug (diameter d=1,5cm; length l=3cm; mass m=5g). Before closing the test tube, V'=1ml of water is poured into it. Room temperature is 20^oC , pressure p=10⁵ Pa , relative humidity is σ=20%. When the temperature in the test tube becomes t'=40^oC the plug is shot. I need to find how much the escaping gas has increased the speed of the plug after it had been shot.

I know that the initial speed with which the plug starts to move is v=12.134m/s.

Homework Equations

Ideal gas laws.

The Attempt at a Solution

I know that:

V_plug=dπl=14.13 cm³
V_{free space}=30ml-14.13ml=15.87ml
t/V_{free space}=t'/V

What now?[/B]

 

[SteamKing]

Homework Statement

The cannon is made from V=30ml test tube completely closed with a plug (diameter d=1,5cm; length l=3cm; mass m=5g). Before closing the test tube, V'=1ml of water is poured into it. Room temperature is 20^oC , pressure p=10⁵ Pa , relative humidity is σ=20%. When the temperature in the test tube becomes t'=40^oC the plug is shot. I need to find how much the escaping gas has increased the speed of the plug after it had been shot.

I know that the initial speed with which the plug starts to move is v=12.134m/s.

Homework Equations

Ideal gas laws.

The Attempt at a Solution

I know that:

V_plug=dπl=14.13 cm³
V_{free space}=30ml-14.13ml=15.87ml
t/V_{free space}=t'/V

What now?[/B]

Where did you get the formula V_plug = dπl ?

 

[kaspis245]

V=S_surface*length

V=2πr*l=dπl

 

[SteamKing]

V=S_surface*length

V=2πr*l=dπl

That's not the formula for the volume of a cylindrical plug. In general, surface area × length ≠ volume

 

[haruspex]

That's not the formula for the volume of a cylindrical plug. In general, surface area × length ≠ volume

I'm not sure what S_surface is intended to mean, but what kaspis has actually done is circumference x length.

 

[SteamKing]

I'm not sure what S_surface is intended to mean, but what kaspis has actually done is circumference x length.

Whatever. It's not a volume.

 

[kaspis245]

V_plug=(d/2)²π=7.065cm³
V_{free space}=30ml-7.065ml=22.935cm³
Ok?

 

[SteamKing]

V_plug=(d/2)²π=7.065cm³
V_{free space}=30ml-7.065ml=22.935cm³
Ok?

Nope. You still haven't got the correct formula for the volume of a cylinder.

http://en.wikipedia.org/wiki/Cylinder_(geometry)

 

[haruspex]

V_plug=(d/2)²π=7.065cm³
V_{free space}=30ml-7.065ml=22.935cm³
Ok?

Notice how you squared a distance but showed the units as cm cubed. That should be a clue that your formula is wrong..

 

[kaspis245]

V_plug=21.195cm³
V_{free space}=8.805ml

 

[SteamKing]

V_plug=21.195cm³
V_{free space}=8.805ml

No, this is not correct.

The diameter of the plug = 1.5 cm x length = 3 cm.
The capacity of the entire test tube is only 30 ml. According to your calculations, it is taken up mostly by the plug, which is wildly improbable.

In the calculation you made in Post #7, you forgot to divide d by 2 when doing the arithmetic.

 

[kaspis245]

V_plug=5.3ml
V_{free space}=24.7ml

 

[SteamKing]

V_plug=5.3ml
V_{free space}=24.7ml

That's the correct volume for the plug.

 

[kaspis245]

What now?