# ENGINEERING MECHANICS 373

## EXTRA CREDIT PROJECT

### Compiled
by: Allen Roelfs

For: Dr. M. Negahban

December 4, 2002

For my project, I chose to model the collision of three
objects in a horizontal plane. This
example is quite similar to imagining three hockey pucks, all of equal mass, on
a sheet of ice with little or no friction acting between themselves and the
surface of the ice. As the first puck is
hit, it is given an initial velocity of V(o-a) and it inevitably strikes puck
two. How do we relate the final velocity
of puck “a” and the initial velocity of puck “b”?

The answer
lies in a characteristic of each collision known as the coefficient of
restitution. Simply put, the coefficient
of restitution is the amount of “bounciness” there is between two objects in a
collision and tells us how the final velocity of each object will be
determined. The relation between
velocities and the coefficient of restitution is given as

V’a -V’b = e (Va – Vb), were “e” is the
coefficient of restitution and the symbol prime indicates final velocity.

In this example, all blocks have equal mass, and block “A”
has initial velocity 1.5 m/s to the left.
The coefficient of restitution for each collision is shown. The derivation of each initial and final
velocity goes as follows:

MaVa + MbVb = MaV’a + MbV’b

Va = V’a + V’b

1.5 = V’a + V’b

V’a – V’b = e(Vb - Va)

1.5 – V’b – V’b = 0.8(-1.5)

-2(V’b) + 1.5 = -1.2

2.7 = -2(V’b)

#### V’b (initial) = 1.35 m/s

V’c = 1.5 – 1.35

#### V’a = 0.15 m/s

Vb
+ Vc = V’b + V’c

Vb
= V’b + V’c

1.35
= V’b +V’c

V’b
– V’c = e (Vc – Vb)

1.35
– V’c – V’c = 0.5 (-1.35)

1.35
– 2(V’c) = -.675

2.025
= 2 (V’c)

#### V’c = 1.0125 m/s

** **

##### V’b = 1.35 – 1.0125

###### V’b (final) =
0.3375 m/s

The
results of this derivation are confirmed by examining the working
model

and
observing the velocity indicators on the page.

To give a better idea of a slightly
more practical application of the

coefficient of restitution, I have
prepared another working model of a collision between a truck and a tree
occurring at excessive highway speeds.
In this simulation, you will notice how the truck responds with very
little elasticity to the tree. This is
due to the low value of the coefficient of restitution due to the composition
of the truck, being mostly steel. To
view this simulation, click here.

I
hope you have found this short explanation of momentum conservation and the
effects of the coefficient of restitution on colliding particles useful in your
study of dynamics.

Allen Roelfs