PH 101 MOMENTUM + COLLISIONS
Momentum and Collisions Notes
General Overview
The lecture covers the topic of momentum in PH101, focusing on its conservation and key principles.
Conservation of Energy has been previously discussed; this lecture shifts focus to momentum, another conserved quantity that can transfer between objects through interactions.
Outline of Key Concepts
Linear Momentum
System of Particles
Center of Mass
Internal Interactions
1D Collisions
Totally Inelastic Collisions
Inelastic Collisions
Elastic Collisions
Linear Momentum
The linear momentum of a single particle is defined as:
ar{p} = mar{v}
Where m is mass and v is velocity.
Momentum is a vector quantity, which means it has both magnitude and direction, pointing in the direction of the velocity.
Units of linear momentum are given as:
[p] = ext{kg m/s}
Angular momentum will also be discussed later; without qualifiers, momentum usually refers to linear momentum.
Momentum and Kinetic Energy
The relationship between kinetic energy (K) and momentum (p) is expressed as:
K = rac{p^2}{2m}
Rearranging gives:
p = ext{√}(2mK)
Key relationships to note:
For two objects with the same mass, the one with more momentum has more kinetic energy.
Conversely, the object with less momentum has less kinetic energy.
For objects with the same momentum, the object with more mass has less kinetic energy.
The one with less mass has more kinetic energy.
For objects with the same kinetic energy, the object with more mass has more momentum while the one with less mass has less momentum.
Change in Momentum
If mass is constant, the rate of change of momentum is equal to net force as described by:
rac{ar{ ext{Δ}p}}{ ext{Δ}t} = m rac{ar{ ext{Δ}v}}{ ext{Δ}t} = mar{a} = ar{F}_{ ext{net}}
This indicates the original form of Newton's 2nd Law.
If net force is zero, momentum remains constant.
The impulse experienced by an object is defined as:
ar{J} = ar{F}dt = ar{Δp}
This corresponds to the area under the force versus time curve.
For a finite force acting over time, average force can be calculated as:
ar{F}_{ ext{av}} = rac{ar{Δp}}{ ext{Δ}t}
Impulse and Work
Impulse is the change in momentum due to a force acting over time:
ar{Δp} = ar{J}_{ ext{net}}
For three dimensions, it can be expressed as:
ar{Δpx} = ar{J}{ ext{net},x}
ar{Δpy} = ar{J}{ ext{net},y}
ar{Δpz} = ar{J}{ ext{net},z}
Work is defined as the force acting over a distance, leading to a change in energy:
ar{ΔE} = ar{W}_{ ext{net}}
Total Momentum of a System of Particles
The momentum of a system consisting of multiple particles is the sum of individual momenta:
ar{P} = ext{Σ}{i} ar{p}i
For a system of two particles (1 and 2):
Total momentum:
ar{P} = ar{p}1 + ar{p}2
The rate of change of total momentum with respect to time can be expressed as:
rac{ar{ΔP}}{ ext{Δ}t} = rac{ar{Δp1}}{ ext{Δ}t} + rac{ar{Δp2}}{ ext{Δ}t}
This can be decomposed into net external forces acting on each particle and the interaction forces.
Isolated Systems
In an isolated system, both total momentum and total energy are conserved:
ar{ΔP} = ar{J}_{ ext{ext}}
ar{ΔE} = ar{W}_{ ext{ext}}
For isolated systems, external impulse and work are zero:
ar{ΔP} = 0
ar{ΔE} = 0
Both total momentum and total energy of an isolated system are conserved, but note that total kinetic energy may or may not be conserved.
Center of Mass
The center of mass (ar{r}_{cm}) of a system of particles is determined as the weighted average of their positions:
ar{r}{cm} = rac{Σ{i} mi ar{r}i}{Σ{i} mi}
The velocity of the center of mass is:
ar{v}{cm} = rac{Δar{r}{cm}}{Δt} = rac{1}{M} Σ{i} mi ar{v}i = rac{1}{M} Σ{i} ar{p}_i
Here, M is the total mass of the system.
The total momentum is presented as:
ar{P} = Mar{v}_{cm}
Motion of the Center of Mass
The acceleration of the center of mass (ar{a}_{cm}) is the time derivative of velocity:
ar{a}{cm} = rac{Δar{v}{cm}}{Δt} = rac{1}{M} rac{Δar{P}}{Δt} = rac{ar{F}_{ ext{ext}}}{M}
This defines the relationship as:
The acceleration of the center of mass is proportional to the net external force:
ar{F}{ ext{ext}} = Mar{a}{cm}
In isolated systems:
If ar{F}_{ ext{ext}} = 0, then the center of mass moves with a constant velocity.
Internal Interactions
Internal forces can transfer energy and momentum within isolated systems but do not alter total momentum and energy.
Types of Internal Forces:
Gravitational force
Spring force
Electrostatic force
Contact forces (Normal, Friction, Tension)
2-Body Interactions
The discussion centers on isolated systems of two interacting particles only through collisions.
Total momentum conservation can be expressed as:
ar{ΔP} = 0
Considering velocities before and after collision:
m1 ar{v}{1,i} + m2 ar{v}{2,i} = m1 ar{v}{1,f} + m2 ar{v}{2,f}
This implies:
m1 ar{Δv}1 = -m2 ar{Δv}2
ar{Δp}1 = -ar{Δp}2
Change in Kinetic Energy and Coefficient of Restitution
The change in kinetic energy during a collision depends on the coefficient of restitution (e):
e = rac{| ar{v}{2,f} - ar{v}{1,f}|}{| ar{v}{2,i} - ar{v}{1,i}|}
Based on the value of e the collision types can be classified as:
Totally Inelastic (e = 0): Total kinetic energy decreases (ΔK < 0).
Inelastic (e < 1): Total kinetic energy decreases (ΔK < 0).
Elastic (e = 1): Total kinetic energy is conserved (ΔK = 0).
Superelastic (e > 1): Total kinetic energy increases (ΔK > 0).
1D Collisions Conservation Principle
In one-dimensional collisions, conservation of momentum yields:
ar{ΔPx} = 0, ar{ΔPy} = 0, ar{ΔP_z} = 0
If they are one-dimensional (along x-axis):
ar{P}x = ext{constant}, ar{P}y = 0, ar{P}_z = 0
Final Velocities from 1D Collisions
Final velocities of two objects can be expressed in terms of their masses, initial velocities, and coefficient of restitution:
ar{v}{1,f} = rac{1}{M} [m1 v{1,i} + m2 v{2,i} + e m2 (v{2,i} - v{1,i})]
ar{v}{2,f} = rac{1}{M} [m1 v{1,i} + m2 v{2,i} - e m1 (v{2,i} - v{1,i})]
The coefficient of restitution is often undetermined except for special cases: e = 0 (perfectly inelastic) and e = 1 (perfectly elastic).
Totally Inelastic Collisions
A totally inelastic collision occurs when two objects stick together after collision, resulting in zero relative motion (e = 0):
ar{v}{2,f} = ar{v}{1,f}
Total momentum before collision is given by:
ar{P}{i} = m1 v{1,i} + m2 v_{2,i}
Total momentum after collision is:
ar{P}{f} = (m1 + m2) vf
Final velocity of combined mass:
vf = rac{1}{M} (m1 v{1,i} + m2 v{2,i}) = rac{ar{P}}{M} = ar{v}{cm}
Inelastic Collisions
In handling general inelastic collisions:
Initial momentum:
ar{P}i = m1 v{1,i} + m2 v_{2,i}
Final momentum:
ar{P}f = m1 v{1,f} + m2 v_{2,f}
There are 6 variables with only one condition (conservation of momentum): ar{P}f = ar{P}i.
Elastic Collisions
Elastic collisions conserve both momentum and kinetic energy:
Initial momentum:
ar{P}i = m1 v{1,i} + m2 v_{2,i}
Initial kinetic energy:
Ki = rac{1}{2} m1 v{1,i}^2 + rac{1}{2} m2 v_{2,i}^2
Final momentum:
ar{P}f = m1 v{1,f} + m2 v_{2,f}
Final kinetic energy:
Kf = rac{1}{2} m1 v{1,f}^2 + rac{1}{2} m2 v_{2,f}^2
The final velocities are given by:
v{1,f} = rac{1}{M} [(m1 - m2)v{1,i} + 2m2v{2,i}]
v{2,f} = rac{1}{M} [(m2 - m1)v{2,i} + 2m1v{1,i}]
Special Cases of Elastic Collisions
Equal masses ($m1 = m2$):
v{1,f} = v{2,i}
v{2,f} = v{1,i}
Very unequal masses ($m1 ext{≫} m2$):
v{1,f} ≈ v{1,i}
v{2,f} ≈ 2v{1,i} - v_{2,i}
Stationary target ($v_{2,i} = 0$):
v{1,f} = rac{m1 - m2}{M} v{1,i}
v{2,f} = rac{2m1}{M} v_{1,i}
Superelastic Collisions
In a superelastic collision, the total kinetic energy of the system increases, which must correspond to a decrease in some other form of energy, such as:
Gravitational potential energy
Elastic potential energy
Chemical energy
Rest energy
Nuclear energy
Closing Remarks
For an isolated system, key points include:
The total momentum remains constant.
The center of mass moves with a constant velocity.
Momentum can be exchanged between parts through internal interactions (like collisions).
The conservation of total kinetic energy is not guaranteed; it depends on the nature of the internal interactions.