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