Download Lecture 8 - Center for Solar-Terrestrial Research

Physics 111: Elementary
Mechanics – Lecture 8
Carsten Denker
NJIT Physics Department
Center for Solar–Terrestrial Research
Introduction
Collisions
 Impulse and Linear Momentum



Single Collision
Series of Collisions
Momentum and Kinetic Energy
 Inelastic Collisions in One Dimension



One-Dimensional Collision
Completely Inelastic Collision
Elastic Collisions in One Dimension
 Collisions in Two Dimensions

October 24, 2006
Center for Solar-Terrestrial Research
Center of Mass for a System of Particles
The center of mass of a body or a system of bodies moves
as though all of the mass were concentrated there and all
external forces were applied there.
xcom
m1 x1  m2 x2

m1  m2
2 bodies, 1 dimension
n bodies, 3 dimensions
xcom
1

M
rcom
1

M
n
m x ,
i 1
i i
n
m r
i 1
October 24, 2006
i i
ycom
1

M
n
m y
i 1
i
i
and zcom
1

M
n
m z
i 1
i i
n bodies, 3 dimensions, vector equation
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Center of Mass for a Solid Body
xcom 
1
M
 xdm, ycom 
1
M
 ydm and zcom 
1
M
 zdm
Differential mass element dm
Uniform density

dm M

dV V
xcom
1
1
1
  xdV , ycom   ydV and zcom   zdV
V
V
V
October 24, 2006
Center for Solar-Terrestrial Research
Newton’s 2nd Law for a
System of Particles
System of particles
Fnet  Macom
A firework rocket explodes
A grand jeté
October 24, 2006
Center for Solar-Terrestrial Research
Linear Momentum
Particle
p  mv
System
Fnet
dp

dt
P  Mvcom
dP
Fnet 
dt
Conservation of Linear Momentum
P  const.  Pi  Pf
If no net external force acts on a system of particles, the total linear
momentum P of the system cannot change.
If the component of the net external force on a closed system is zero
along an axis, then the component of the linear momentum along that
axis cannot change.
October 24, 2006
Center for Solar-Terrestrial Research
Impulse and Linear Momentum
Definition of Impulse
dp  F  t  dt

pf
pi
Collision of two
particle-like bodies
dp   F  t  dt
tf
ti
J   F  t  dt
tf
ti
Impulse–Linear
Momentum Theorem
p f  pi  p  J
Steady stream of projectiles
October 24, 2006
Center for Solar-Terrestrial Research
Momentum and Kinetic Energy
Closed system (no mass enters or leaves)
 Isolated system (no external net force)
 Elastic collision (kinetic energy conserved)
 Inelastic collision (kinetic energy not conserved)
 Completely inelastic collision (bodies always stick
together)

In a closed, isolated system containing a collision, the linear
momentum of each colliding body may change but the total
momentum P of the system cannot change, whether the
collision is elastic or inelastic.
October 24, 2006
Center for Solar-Terrestrial Research
Inelastic Collisions in 1D
Conservation of Linear Momentum
p1i  p2 f  p1 f  p2 f
Completely Inelastic Collision
m1v1i   m1  m2  v
v
m1
vi
m1  m2
Velocity of Center of Mass
P  Mvcom   m1  m2  vcom
vcom
p1i  p2i
P


m1  m2 m1  m2
October 24, 2006
Center for Solar-Terrestrial Research
Elastic Collisions in 1D
In an elastic collision, the kinetic energy of each colliding body may
change, but the total kinetic energy of the system does not change.
Stationary Target
m1v1i  m1v1 f  m2v2 f
1
1
1
m1v12i  m1v12f  m2 v22 f
2
2
2
m  m2
v1 f  1
v1i
m1  m2
v2 f 
2m1
v1i
m1  m2
October 24, 2006
Moving Target
m1v1i  m2 v2i  m1v1 f  m2v2 f
1
1
1
1
m1v12i  m2v22i  m1v12f  m2v22 f
2
2
2
2
m  m2
2m2
v1 f  1
v1i 
v2i
m1  m2
m1  m2
v2 f 
2m1
m  m1
v1i  2
v2i
m1  m2
m1  m2
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