Why Whips Crack
Why Whips Crack
Why Whips Crack
(revised 9/11)
Copyright Ambrose McNibble 2005-2011
What makes a whip crack is the tip exceeding the speed of sound. The crack is a
miniature sonic boom just like the one an airplane creates when it exceeds the
speed of sound. The speed of sound at sea level is approximately 760 miles per
hour or about 340 meters per second (it varies a few miles per hour depending on
temperature and air pressure) Some people have trouble believing that physics
would allow a person - even a very strong person - to throw anything at 760 miles
per hour. Since it's easily demonstrable that even a child can crack a whip, these
people believe something else must be making the soundperhaps the whip is
hitting itself. Contrary to this belief, physics shows that a sonic boom is exactly
what happens.
I will use three equations, some simple algebra and the concept of ratios to show
this.
Kinetic Energy
Kinetic energy is the energy of motion. The kinetic energy of an object is the
energy it possesses because of its motion. Kinetic energy is an expression of the
fact that a moving object can do work on anything it hits; it quantifies the amount
of work the object could do as a result of its motion. The kinetic energy of a point
mass m is given by:
equation 1
E = 12 m(V 2 )
V=
2E
m
Where:
E = energy in Joules
m = mass in grams
V = velocity in meters per second
Ratios
We all know that 1/2 is more than 1/100, and that is the key to understanding what
happens when a whip cracks. When a whip is thrown, the part that's moving, the
part that 'holds' the kinetic energy shrinks, but the energy remains about the same.
This is represented in equation 4b by the m in the bottom-right of the equation. As
m gets smaller, the number represented by the equation, V, gets bigger. This
manifests as a dramatic increase in the speed of the tip as a whip approaches full
extension.
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This is perhaps intimidating, but you're past the worst of it, and I did say I was
going to use equations and ratios. I promise to make it as easy as I can. I have put
the algebraic translation of the above equation at the end so mathphobes will not be
too bothered.
Given that professional baseball players routinely throw baseballs at over 90 miles
per hour, 25 miles per hour is a reasonable assumption for how fast a whip can be
thrown.
For this discussion, let us assume the part of the whip that is moving has a mass of
6.2oz, or about 175 grams.
If we put these numbers into equation 1, we arrive at:
equation 1
E = 12 m(V 2 )
In the real world, some energy will be lost to internal friction as the bend moves
down the whip and to resistance by the air to the moving whip. For this discussion,
let us assume that 15% of the energy is lost and the other 85% of the energy is
transferred to, and concentrated in, the moving part of the whip. Taking our energy
number from above, we have:
Let us assume the part of the whip that is moving is just the cracker, and it has a
mass of five one-thousandths (.005) of an ounce, or about 0.15 grams.
If we put these numbers into equation 4b, we arrive at:
equation 4b
V=
V=
2E
m
2 9,815 140
=
= 362meters /sec
.387
.15
362 meters per second works out to about 810 miles per hour - well above the 760
mph required to create a sonic boom.
In practice, as the bend in the whip approaches the tip, the mass of the moving part
of the whip approaches zero, so we could pick any small number for the mass of
the tip that makes our equations work, and simply wait a fraction of a second until
that was the only part of the whip left moving.
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The conclusion that whips exceed the speed of sound is born out by
shadowgraphsphotographs of the shadows caused by the refraction of light by
the shock waves of a whip's sonic booms. The first of these were taken in 1927,
although the correct interpretation was not applied until 1958 in a paper by B.
Bernstein, D. A. Hall, and H. M. Trent of the U.S. Navel Laboratory.
Shadowgraphs from whip cracks can be found several places on the Internet.
An example of the second mode of motion I identified occurs when the whip is
held with the handle horizontal, and the handle is moved up and down vigorously a
short distance - no more than ten inches. This causes a wave shape or 'S-curve' to
form and move down the whip carrying the energy from the handle to the tip.
When the wave reaches the tip, the physics is the same as with the simple overhand
throw. This is a 'ringmaster's crack' done with restrained movement. The energy
arrives at the cracker as a full-wave motion rather than linear motion as in the
overhand throw.
This throw can be done in any plane of rotation as long as the movement of the
handle is perpendicular to the body of the whip. Done correctly, the whip will
crack twice in rapid successiononce for each reversal of direction.
The third mode of motion I have identified occurs when a whip is thrown in a way
that forms a loop that moves down the whip carrying energy from the handle to the
tip. When the loop reaches the tip of the whip, it unrolls 180 and assumes the halfwave configuration discussed above, and the same physics apply as in that mode of
motion. I do not yet have simple instructions for producing an example of this
mode.
The energy arrives at the cracker as both rotary and liner motion.
Although Professor A. Goriely does not mention modes of motion in his paper, I
believe this is the mode he was looking at in his first analysis of whip cracking.
This mode may be easier to achieve with longer whips.
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2E = 2 12 m(V 2 )
2E = m(V 2 )
2E m (V 2 )
= m
m
The m's on the right cancel giving us:
equation 3b
2E
=V2
m
2E
=V
m
Or the equivalent equation:
equation 4b
V=
2E
m
The key when looking at equation 4b is that the term m for mass is on the bottom
of the fraction on the right side and the term V for velocity is on the top on the left.
This gives them an inverse relationship such that when one approaches zero, the
other approaches infinity.
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equation 5
V2 =
m1 0.85
V1
m2
Where:
m1 = the mass of the whip body in grams
m2 = the mass of the whip cracker in grams
V1 = the velocity the whip is thrown in meters/sec
V2 = terminal velocity of the cracker in meters/sec
I leave it as an exercise for the reader to convert these numbers to other units if
they desire to. The conversion factors are all readily available to anyone with a
computer.
It can be amusing and enlightening to create a spreadsheet and play with the
variables in this formula. It will rapidly become clear why longer and/or heaver
whips are easier to crack, and why thin crackers with few strands are easier to
crack than fat, fluffy crackers with many strands.
Bottom line is: long heavy whips with thin crackers will get more of their length
above the speed of sound and make louder cracks or crack with less effort or less
skill.
What is perhaps not so clear from these calculations is how these variables effect
accuracy and what happens when something other than 'perfect' throws are made.
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