Page 180 - Standard Handbook Of Petroleum & Natural Gas Engineering

Dynamics 165

Note that e is defined in terms of the components of the velocities, not the vector
velocities, whereas the momentum balance is defined in terms of the vector
velocities. To solve Equations 2-32 and 2-33 when all the velocities are not colinear,
one writes the momentum balances along the principal axes and solves the resulting
equations simultaneously.
For purely elastic impacts, e = 1, and for purely plastic impacts, e = 0. For elasto-
plastic impacts, e lies between zero and one and is a function of both the material
properties and the velocity of impact.
Example 2-1 0

Sphere 1 weighs 1 Ib and is traveling at 2 ft/s in the positive x direction when it
strikes sphere 2, weighing 5 lb and traveling in the negative x direction at 1 ft/s.
What will be the final velocity of the system if the collision is (a) plastic, or (b)
Elastoplastic with e = 0.5!
(a) By Equation 2-32

mlVll - m,V,, - 2 - 5
V= - -0.5 ft/s
m, +my 6
(b) By Equation 2-34

v,, = vI2 + 0.5[2 - (-1)] = v,, + 1.5

By Equation 2-33

mlvl, + m,v,, = m,vI2 + m,v,, = m,vl2 + m,(V,, + 1.5)
= (m, + m2)v,, + 1.5m2

mlv, +m,(v, - 1.5) 1 x 2 +5(-1 -1.5)
v,, = - = -1.75ft/s
-
m, +my 1+5
v,, = -0.25 ft/s

The foregoing discussion of impulse and momentum applies only when no change
in rotational motion is involved. There is an analogous set of equations for angular
impulse and impulse momentum. The angular momentum about an axis through the
center of mass is defined as
-
H = f~ (2-35)
and the angular momentum about any arbitrary point 0 is defined as

H, = lo + mGd (2-36)