Post on 15-Jan-2016
Physics 111: Elementary Mechanics – Lecture 8
Carsten Denker
NJIT Physics DepartmentCenter for Solar–Terrestrial
Research
October 24, 2006 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.
2 bodies, 1 dimension
n bodies, 3 dimensions
n bodies, 3 dimensions, vector equation
1 1 2 2com
1 2
m x m xx
m m
com com com1 1 1
1 1 1, y and
n n n
i i i i i ii i i
x m x m y z m zM M M
1
1 n
com i ii
r m rM
October 24, 2006 Center for Solar-Terrestrial Research
Center of Mass for a Solid Body
Differential mass element dm
Uniform density
com com com
1 1 1, and x xdm y ydm z zdm
M M M
dm M
dV V
com com com
1 1 1, and x xdV y ydV z zdV
V V V
October 24, 2006 Center for Solar-Terrestrial Research
Newton’s 2nd Law for a
System of Particles
A firework rocket explodes
A grand jeté
System of particles
net comF Ma
October 24, 2006 Center for Solar-Terrestrial Research
Linear MomentumParticle System
Conservation of Linear Momentum
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.
p mv
net
dpF
dt
comP Mv
net
dPF
dt
const. i fP P P
October 24, 2006 Center for Solar-Terrestrial Research
Impulse and Linear Momentum
Definition of Impulse
Impulse–Linear Momentum Theorem
Collision of two particle-like bodies
Steady stream of projectiles
f f
i i
f
i
p t
p t
t
t
dp F t dt
dp F t dt
J F t dt
f ip p p J
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 of the system cannot change, whether the collision is elastic or inelastic..
P
October 24, 2006 Center for Solar-Terrestrial Research
Inelastic Collisions in 1DConservation of Linear Momentum
Completely Inelastic Collision
Velocity of Center of Mass
1 2 1 2i f f fp p p p
1 1 1 2
1
1 2
i
i
m v m m v
mv vm m
com 1 2 com
1 2com
1 2 1 2
i i
P Mv m m v
p pPv
m m m m
October 24, 2006 Center for Solar-Terrestrial Research
Elastic Collisions in 1D
Stationary Target Moving Target
In an elastic collision, the kinetic energy of each colliding body may change, but the total kinetic energy of the system does not change.
1 1 1 1 2 2
2 2 21 1 1 1 2 2
1 21 1
1 2
12 1
1 2
1 1 1
2 2 2
2
i f f
i f f
f i
f i
m v m v m v
m v m v m v
m mv v
m m
mv v
m m
1 1 2 2 1 1 2 2
2 2 2 21 1 2 2 1 1 2 2
1 2 21 1 2
1 2 1 2
1 2 12 1 2
1 2 1 2
1 1 1 1
2 2 2 22
2
i i f f
i i f f
f i i
f i i
m v m v m v m v
m v m v m v m v
m m mv v v
m m m m
m m mv v v
m m m m