Tuesday, April 3, 2012

Pies in the Face and Inelastic Collisions

Welcome to blog entry # 68 of my math and calculator blog!

Before I start, I dedicate this blog entry to pranksters, professional clowns (the ones whose job is it to bring laughs, not the annoying politician kind), Shane Jensen, Chris Brame, A.J. Burnett, the Texas Rangers, and the New York Yankees.

Today I am on the tail end of a nasty cold. The saying "laughter is the best medicine" and the beginning of the 2012 MLB baseball season reminds me of the good old "pie in the face" prank. Yes, I like baseball for a lot of reasons and I have grown to like the pieing prank as part of it.

This blog entry, I attempt to bring math, physics, and cream pies together. For those of you who expect a serious subject, my apologizes, but I am going for the silly side today.

Here is what typically happens when someone throws a pie in... a face. Disclaimer: I do not claim to be an artist by any means.

So I break the pie in the face into two stages:

1. Before to Splat
2. Splat to Aftermath

Since the pie sticks to the receiver's head, it is safe to say that the pie in the face row is an example of an inelastic collision. Momentum will be preserved but not kinetic energy.


m_pie = mass of a cream pie, about 1 to 2 pounds

m_arm = mass of the thrower's arm, which is about 6.5% of a human's weight according to Shirley A Kindrick, PhD of The Ohio State University.

m_head = mass of the receiver's head. It is approximately 8 to 12 pounds according to The Physics Factbook edited by Glenn Elert.

v_t = velocity of the arm throwing the pie

v_f = velocity of the arm, pie, and head after the contact

v_a = velocity of the pie covered head after contact


1. The entire pie in the face motion is done in one-dimension. That is, back and forth. The receiver is getting the pie head on at no angle.

2. In the Splat to Aftermath stage, the thrower's arm "falls" out of the picture, allowing the reciever's head to come back to neutral.

3. The amount of "pie" lost is negligible when the contact occurs.

4. To keep is problem simple, the thrower does not "smear" the pie over the receiver.

5. Both the thrower and receiver are standing still. The receiver is not trying to duck, avoid, or do anything to avoid being pied. The receiver exerts no effort after being pied.

Momentum Conservation

Before to Splat Stage:

(m_pie + m_arm) * v_t = (m_pie + m_arm + m_head) * v_f

Splat to Aftermath Stage:

(m_pie + m_arm + m_head) * v_f = (m_pie + m_head) * -v_a

v_a is negative since the head is moving "back" after the thrower releases his grip.

Let's try a Numeric Example

I am going to use US Customary units here:

* pounds for mass (lb) [the SI unit is kg]
* feet/second for velocity (ft/s) [the SI unit is m/s]
* foot-pound (lb_f) for a unit of energy [the SI unit is J (Joule)]

Conversion: 1 mi/hr = 22/15 ft/s. (approximately 1.466667)

A thrower, weighing 210 pounds is holding a 1.5 pound pie at 0.6 mph. He launches it on to his friend's face. His friends weights 220 pounds, assuming his head weights 10 pounds. What is the velocity of friends's head at contact and after contact? What is the kinetic energy at the competition of the two stages?

m_pie = 1.5 lb
m_arm = 6.5% * 210 = 13.65 lbs
m_head = 10 lbs
v_t = 0.6 mph = 0.88 ft/s

Before to Splat Stage:
(1.5 + 13.65) * 0.88 = (1.5 + 13.65 + 10) * v_f
15.15 * 0.88 = 25.15 * v_f
13.33202 = 25.15 v_f
0.53010 = v_f

The arm, pie, and head are moving approximately 0.53010 ft/s (0.77748 mph) at Splat.

Kinetic Energy (KE = 1/2 * mass * velocity^2)
KE before Splat = 1/2 * 15.15 * 0.88^2 ≈ 5.86608 ft_lb
KE at Splat = 1/2 * 25.15 * .53010^2 ≈ 3.53365 ft_lb
(a loss of approximately 39.8%)

Splat to Aftermath Stage:
25.15 * .53010 = (1.5 + 10) * -v_a
13.33202 = 11.5 * -v_a
v_a ≈ -1.15931

At the aftermath stage, the velocity of the pied head is 1.15931 ft/s (0.79044 mph) in the other direction, a small whiplash after pushing the friend back with that pie.

Kinetic Energy:
At Aftermath: KE = 1/2 * 11.5 * 1.15931^2 ≈ 7.728 lb_ft
(more than double after release due to loss of mass)

So this is an attempt to explain the physics of the pie in the face.

Thanks for reading as always. To Shane Jansen, Chris Brame and those who like pies in the face, I splat you with a virtual pie.

This blog is property of Edward Shore. © 2012

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