Lawrence Wu
Biology 438
April 15th, 2014

Act of raising one's center of gravity higher in
the vertical plane solely with the use of one's
own muscles
Lees et al., 2004 Journal
of Biomechanics
http://www.calworkouts.com/images/lifti
ng/Muscles_anterior_upper_labeled.png
http://www.verticaljumpcenter.com/wpcontent/uploads/2012/06/legmuscles.jpg

Fast twitch muscles
◦ Overall: Produce great amount of work in small
amount of time
◦ Optimal balance between high velocity of
contraction and force generated to maximize power
◦ Knees flex and muscles become increasingly stiff
 Body halts the fast stretch of the muscle
 Great amount of force produced in the muscle and a
rise in elastic energy storage in the muscles

Analyze components of vertical jump
◦ Muscles involved
◦ Physics
 Elasticity
 Rotational motion
 1D Kinematics

How does the counterswing arm movement
affect height of jump?
◦ Other factors involved
 Power, energy

Calculated center of mass based on average
segment weight for different parts of body
◦ http://www.exrx.net/Kinesiology/Segments.html

No arm movement
◦ Knee, Foot, Hip
◦ Model: mass (upper body) on spring (legs)

Arm movement
◦ Tracked movement of
 Knee, Foot, Hip, Shoulder, Elbow
◦ Model: mass (upper body) on spring (legs) with a
force applied upwards (arm movement)
Arm movement
(“External force”)
Upper body
(Mass)
Legs (Spring)

video
*Simple harmonic motion
compression
(+) restoration
(-) restoration
Minor Compression
And restoration
compression
(+) restoration
(-) restoration
Minor Compression
And restoration

Apparent “spring constant” of legs: 1465 N/m
◦ Ei=Ef= ∆ Work =345Joules
◦ Assumption: PEelastic,legs=PEgravity  0.5kx2=mgh
◦ x=0.95-0.26m=0.69m (change in ycm of whole body during
compression)
◦ m=68.04 kg, h=0.52m (change in ycm of whole body after
restoration))
Rest
Compression
Restoration

Average Power: 495.7 Watts
◦ 𝑃=
Work
∆𝑡
 Work=345 Joules
 ∆𝑡=0.696 s

Work generated by contraction of muscle
groups in legs

video
*Simple harmonic motion
compression
(-) restoration (+) restoration
Minor Compression
And restoration
compression
(-) restoration (+) restoration
Minor Compression
And restoration

Ei=Ef=∆ Work= 543 Joules
◦ Assumption: PEelastic, legs+KE rotational=PEgravity
◦ 0.5kx2+ 0.5I 𝝎 2 =mgh
◦ k=1465 N/m, h=.81m, x=.54m
Rest
Compression
*Estimated center
of mass since arms
are not visible
Restoration

Rotational energy
1
3
1
3
◦ Iarm= MforearmR2forearm+ Mupper armR2upper arm+MforearmR2forearm
◦ Iarm=.19 kg m2
∆𝜃
∆𝑡
◦ 𝜔= , ∆𝜃=5.236 radians, ∆t=.472, 𝜔= 11.09 rad/s




◦ KErotational=11.7 Joules
PEelastic, legs+KE rotational≠PEgravity
New equation: PEelastic, legs+KE rotational+ PEelastic, arms to legs=PEgravity
PEelastic, arms to legs=317.7 Joules
New “apparent spring constant”



.5kx2+.5I 𝜔 2=mgh
.5kx2=543 J-11.7 Joules
K=3644 N/m

Average Power: 742 Watts
◦ 𝑃=
Work
∆𝑡
 Work=543 Joules
 ∆𝑡=0.732 s
◦ Average power of arm movement: 24.8 Watts
 Workarm= 11.7 Joules
 ∆𝑡𝑎𝑟𝑚=0.472 s
*Not maximum height of center of mass
No Arm movement (0.53m)
Arm movement (1.10 m)

Arm movement contributes significantly to
the height of the vertical jump (1.10m to
.52m)
◦ Energy as well
 The time of jump with arm movement, from bottom to
maximum height, is only 5% longer (0.732s to 0.696s)
 However arm movement increases the energy of
system by 57% (345 Joules to 543 Joules)
 Mainly potential energy (96% of increased energy)
 Some rotational kinetic energy (4% of increased energy)

Center of mass is accurately modeled
◦ Same expected trajectory as a point particle

Full body movement is most efficient and
produces most power


Recruits additional muscle groups above the torso to
increase force generation
Body-leg spring system is complex

Can be modeled with relatively simple mechanics

Decompose my mass-spring model into
smaller systems to analyze in greater detail
◦ Specific sections of the leg
 Calves, feet, etc.


Analyze how the positioning of the arm adds
potential energy to the whole body system
Electromyogram to analyze individual muscle
activity in real time