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Coefficient of Restitution

Q4E Case Study 3 – The Coefficient of Restitution
Proposed Subject usage: Mathematics / Physics (A/AS level) Sports Science (1st/2nd year)
Newton’s law of restitution. Applications to direct impact of two particles and normal impact of a particle with a plane surface.

Knowledge of the terms perfectly elastic (1 = e) and perfectly inelastic/plastic (0 = e) is expected.

Coefficient of Restitution

Direct impacts of two objects occur when their velocity vectors are parallel. Linear momentum is conserved but energy is lost during the impact when one or both of the objects are deformed. The greater the deformation that occurs the longer the impact lasts. The strain energy stored during the deformation will partly be regained in the latter part of the impact depending on how plastic or elastic the impact is. Newton’s empirical law demonstrates this:

u1 – u2 = e(v1 –v2)

Where:

u  represents the approach velocity
v  is the separation velocity
1 and 2 represent the two objects
e = a constant known as the coefficient of restitution

Method 1- Using velocities to determine e

If the impact is between an object (e.g. a tennis ball) and a surface then there is no velocity prior to or after impact for object 2.

Therefore: as u2 = v2 = 0 in the above equation : v/u = e

Method 2 – Using heights to determine e

v2 = u2 + 2as (one equation of uniform motion) Where: a = gravity (g) s = distance (in this case height)

Therefore: u = v2ghd : v = v2ghb

Where: hd = height dropped (prior to impact) hb = height bounced (after impact)

By substituting u = v2ghd and v = v2ghb into v/u = e

e =v(hb / hd )

This law indicates how e and velocity before impact affects movement after impact. The value of e can range from 0 – 1 with 0 being a perfectly plastic impact and 1 being a perfectly elastic impact.

Methods
Five different sports balls dropped onto a wooden table and filmed at high speed (100fps Basler). The videos were digitised and the data were exported into excel files. Heights of the balls were determined using the distance measuring tool within Quintic. Users can calculate distance by using the line / ruler function). Both methods of calculating e (using bounce height and velocity) can be used and compared.

Functions of the Quintic Software used:

  • Digitisation module
  • Calibration
  • Export data
  • Line and distance measuring tool
  • Split screen
  • On-screen point coordinates

Results
The results shown in the tables below indicate that there is a distinct difference between the impacts of the different balls onto the surface.

Method 1
Table Tennis
Ball

Marble

 

Cricket
Ball
Tennis
Ball
Golf
Ball
Velocity prior to impact (u) (m/s)
1.92
1.55
1.39
2.09
1.66
Velocity after impact (v) (m/s)
1.8
0.82
0.6
1.76
1.23
e

0.94
0.53
0.43
0.84
0.74

 

Method 1
Table Tennis
Ball

Marble

 

Cricket
Ball
Tennis
Ball
Golf
Ball
H dropped (m)
0.22
0.16
0.13
0.27
0.16
H bounced (m)
0.18
0.05
0.03
0.18
0.1
Hb/Hd
0.82
0.31
0.23
0.67
0.63
e

0.90
0.56
0.48
0.82
0.79
Coefficient of Restitution | Quintic Sports