The Vacuum Cannon
A Penetrating View Into the Physics
of Pressure, Force, and
Acceleration…
Pressure, Force, and Area
• Pressure is defined by the equation:
P = pressure
P=F
F = force
A
A = area
• The SI units for pressure are Pascals
N
1Pa  1 2
m
Pressure, Force, and Area
101.3kPa=101300N/m2
• Atmospheric Pressure is
• Force vs. Pressure:
- Force is given by Newton’s 2nd Law: F=ma
- Force tells you how an object will accelerate
- Pressure tells you how an object will feel a Force
Example 1
• Shaquille O’Neal has a mass of 150kg and
wears size 22 shoes, which have an area of
approximately 1000cm2 each. A female
with a mass of 60kg wears stilettos with
an area of 1cm2 on each heel and 24 cm2 on
each toe. Calculate the pressure exerted
by each person on the ground. Which Shoe
would you rather have step on your foot?
Example 1…Con’t (Shaq’s Basketball Shoe)
Force of Gravity:
Fg  mg
 9.8 N 
Fg  (150kg )  

 kg 
Fg  1470 N
Area of Shoe:
 1m 
A  1000cm  

 100cm 
A  0.1m 2
2
2
Pressure: (Don't forget there are 2 Shoes!)
F
A
1470 N
P
2  0.1m 2
P
P  7350 Pa
Example 1…Con’t (High Heels)
Force of Gravity:
Fg  mg
 9.8 N 
Fg  (60kg )  

 kg 
Fg  588 N
Area of Shoe:
 1m 
A  (1cm  24cm )  

 100cm 
A  0.0025m 2
2
2
2
Pressure: (Don't forget there are 2 Shoes!)
P
F
A
P
588 N
2  0.0025m 2
P  117600 Pa
Example 1…Con’t
• So who would you rather have step on
your foot?
Shaq’s Basketball Shoe : P=7350Pa
High Heel : P=117600Pa
Atmospheric Pressure… How Much
is it Really?
Atmospheric Pressure is measured as
101300N/m2. Remember what pressure a
60kg woman in high heels applied when
stepping on 25cm2 of you, it was
117600N/m2. This means that atmospheric
pressure is similar to a 60kg woman
standing in high heels on every 25cm2 of
your body.
Now Let’s Take a Look at an
Application… The Ping-Pong Ball
Vacuum Cannon
A Very Important Piece
The First Thought
What does Newton’s
Second Law tell us?
How can we relate
Pressure, Force and, Area?
What does this give us for
Acceleration under
constant pressure?
F
F  ma  a 
m
F  P0 A
F P0 A
a0  
m
m
Let’s Do Some Math
vf ^ 2  vi ^ 2  2a 0 x

2 P 0 Ax
v( x) 
m
m  2.5 g
3 2
A  1.13 10 m
x  1.5m
5
2
P 0  1.013 10 N / m  1atm
• What muzzle velocity do these equations give us?
v( x  1.5m)  370.6 m
• Is this reasonable?
s
Let’s Correct This
As the ping-pong ball moves along the cannon the
air entering not only accelerates the ping-pong ball
but the growing column of air behind it.1 This
gives us a new equation for velocity.
1 

2


x
m


v( x)  v max
1

2

 
xm
 Ax 

A


v max 
P0

  airdensity  1.21 kg
m
3
This gives us a corrected v(x=1.5m)=241.8m/s
Let’s Look at this Graphically
First Thought
Corrected Equation
Speed of Sound
400
350
300
250
200
150
100
50
0
0
0.5
1
1.5
2
2.5
3
How Pressure is Measured
Relationship between Inches of Mercury vs Atmospheric Pressure
35
30
0, 30
0.1, 27
Inches of Hg (inches)
25
20
15
0.5, 15
10
5
0
1, 0
0
0.2
0.4
0.6
Atm ospheric Pressure (atm )
0.8
1
1.2
A Few Fun Facts from Researchers
• The Ek can be higher than that of a .22
caliber handgun.2
• If He-gas is used to accelerate the ball
instead of air velocities as high as 370m/s
have been recorded. 3
Now Who Wants to See it in Action?
Citations
1. American Journal of Physics, Vol. 72, No.
7, pp. 961–963, July 2004
2. The ballistics of many guns can be found
at
http://www.volny.cz/buchtik/Revo/Ballisti
c_Info_komplet.htm
3. The Physics Teacher, Vol. 43, No. 1, pp.
22–25, January 2005