Hyperbolas and the Sonic
BOOM
By
Sonic BOOM
• The sonic BOOM is a shockwave that airplanes
(other objects) produce by “breaking the sound
barrier.”
• This occurs when the objects travels faster than
the speed of sound.
• The speed of sound varies at different elevations,
but at sea level Mach 1 is 761 mph.
• The shock wave travels backwards and outwards
from the plane in a cone that hits the ground in a
hyperbolic curve.
• Sometimes sonic BOOMs can cause structural
damage to houses.
What sound waves looks like..
Infinite cone = a sonic BOOM
Darker = higher pressure
Lighter = lower pressure
History of the Sonic BOOM
• Captain Charles E. Yeager of the US Air Force piloted the
first plane to break the sound barrier in 1947.
• However, whips and bullets have been breaking the sound
barrier for years.
• Sonic BOOMs were discovered to be hyperbolas when
witnesses in a hyperbolic curve recorded hearing the noise
at the same time.
Mathematics of Sonic BOOMs
• Hyperbola- the set of all points (x,y) in a plane, the
difference of whose distances from the foci is a positive
point
• Foci- two distinct fixed points
• Branches- the two disconnected parts of the graph of a
hyperbola
• Vertices- the line through the two foci intersects the
hyperbola at two points
• Transverse Axis- the line segment connecting the vertices
• Center- the midpoint of the transverse axis
• Variable a and b- define the change of x and y relative to
each other
Mathematics of Sonic BOOMs
For when the transverse is horizontal
2
2
( x − h) ( y − k )
−
=1
2
2
a
b
For when the transverse is vertical
2
2
( y − k ) ( x − h)
−
=1
2
2
a
b
x2 y2
− 2 =1
2
a
b
y2 x2
− 2 =1
2
a
b
Example Problem: Find the point V.
Substitute 500 for a and b, which
are equal in this situation.
Substitute 0 for y
because V is on the
x-axis.
y=0
x = −500 ft
V (−500,0)
x2 y 2
− 2 =1
2
a
b
x2
y2
−
=1
2
2
500 500
x2
y2
−
=1
2500 2500
x2
0
−
=1
2500 2500
x2
=1
2500
x 2 = 2500
x = ± 2500
x = ±500
Application: The affects of the sonic BOOM
on wildlife
f ( x, y ) = z
Why I Chose Sonic BOOMs:
BIG noises in the sky that
break windows are a BIG deal.
Works Cited
• http://www.papermag.com/blogs/IndianaJonesDVD_.jpg
• http://images.military.com/pics/BA_Sonic_Boom_opt.jpg
• http://www.nonoise.org/library/animals/11.gif
• Foster, Maggie. Conversation apropos Calculus III. 17 May 2009.
• Foster, Betsy. Conversation apropos Sonic Booms of the 1950s and 1960s.
17 May 2009.
• Henderson, Tom. “The Doppler Effect and Sound Waves.” Glenbrook South
Physics Teachers. 1996. The Physics Classroom. 13 May 2009.
• Hughes-Hallett, Deborah, Andrew M. Gleason, William G. McCallum, et al.
Calculus: Single and Multivariable, 4th Edition. USA: John Wiley and Sons,
Inc., 2005.
• Larson, Ron, Robert Hostetler, and Bruce H. Edwards. Algebra and
Trigonometry: A Graphing Approach, 5th Edition. Boston: Houghton Miffler
Company, 2008.
• Scott, Jeff. “Chuck Yeager & the Sound Barrier.” 28 January 2001. 17 May
2009. <http://www.aerospaceweb.org/question/history/q0011a.shtml>.
• "sonic boom." Encyclopædia Britannica. 2009. Encyclopædia Britannica
Online. 13 May 2009.
<http://www.britannica.com/EBchecked/topic/554486/sonic-boom>.