1. Biomechanics of Vertical Jump
Lawrence Wu
Biology 438
April 15th, 2014

2. Vertical Jump
Act of raising one's center of gravity higher in the vertical plane solely with the use of one's own muscles
A1- Initial standing position.
A2 to A3- You begin descending in to a 1/2 squat position. The quadriceps are being stretched eccentrically (lengthened) and at a fast pace.
A4 to A6- Muscle counter-movement occurs and quadriceps are made to move concentrically (shorten) to send you flying high. During A2 and A3 the knees flex in order to prepare for the jump. At this time the muscles become increasingly stiff. This is a natural way that your body halts the fast stretch of the muscle. The results of this are a great amount of force being produced in the muscle and a rise in elastic energy storage in the muscles. Also, the faster this sequence is performed, the greater the jump. Many people have conducted experiments to see what benefits and what ails vertical jump performance.
Lees et al., 2004 Journal of Biomechanics

3. Muscles involved
Hip extension
Knee extension
Ankle plantarflexion

4. Muscles Involved 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

5. The Study 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

6. Methods
Calculated center of mass based on average segment weight for different parts of body
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)

7. Simplified model Arm movement (“External force”) Upper body (Mass)
Legs (Spring)

8. No arm movement
video
.584 s

9. Movement of center of mass
*Simple harmonic motion
Label graph: bending of knees (compression), restoration,
Not ideal spring: lost to surroundings (Simple harmonic motion)
compression
(+) restoration
(-) restoration
Minor Compression
And restoration

10. Movement of all modeled body parts
compression
(+) restoration
(-) restoration
Minor Compression
And restoration

11. No arm movement Apparent “spring constant” of legs: 1465 N/m
Ei=Ef= ∆ Work=345Joules
Assumption: PEelastic,legs=PEgravity  0.5kx2=mgh
x= m=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))
Compression
Compression
Restoration
Rest
Rest
Power
T=.732, avgP=

12. No arm movement Average Power: 495.7 Watts
𝑃 = Work ∆𝑡
Work=345 Joules
∆𝑡=0.696 s
Work generated by contraction of muscle groups in legs

13. Arm movement
video

14. Center of Mass movement
*Simple harmonic motion
Calculate g
compression
(-) restoration
(+) restoration
Minor Compression
And restoration

15. Movement of all modeled body parts
compression
(-) restoration
(+) restoration
Minor Compression
And restoration

16. Arm movement 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
*Estimated center of mass since arms are not visible
Compression
Restoration
Rest
.5418
Rotational KE
This energy was used to (i) increase the kinetic and potential energy of the arms at take-off, (ii) store and release energy from the muscles and tendons around the ankle, knee and hip joint, and (iii) ‘pull’ on the body through an upward force acting on the trunk at the shoulder=

17. Arm movement Rotational energy
Iarm= 1 3 MforearmR2forearm+ 1 3 Mupper armR2upper arm+MforearmR2forearm
Iarm=.19 kg m2
𝜔 = ∆𝜃 ∆𝑡 , ∆𝜃=5.236 radians, ∆t=.472, 𝜔 = 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

18. Arm movement 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

19. Difference in Maximum height above ground
*Not maximum height of center of mass
No Arm movement (0.53m)
Arm movement (1.10 m)

20. Conclusions
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)

21. Conclusions 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

22. Future Directions
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