Physics Notes: Linear Momentum and Collisions

Linear Momentum

• Definition: Linear momentum (P) of a mass (m) moving with velocity (v) is given by:
  P  m v

  • Momentum is a vector quantity. The direction of momentum (p) is the same as the direction of the velocity (v).

  • Units: [p] = kg·m/s (no special name assigned).

  • The symbol P for momentum is used due to the unavailability of the letter "m" (already defined for mass).

• Total Momentum:

  • For multiple masses, the total momentum can be expressed as:
    p{tot}  P = P1 + P2 + P3 + … = Sum m v

• Conservation of Momentum:

  • Momentum is conserved in a system isolated from external forces. This principle states:

  • Momentum cannot be created or destroyed, only transferred from one object to another.

  • Implications similar to Conservation of Energy, always true without exceptions.

  • Proofs of momentum conservation to follow later.

• Collision Illustration:

  • For two colliding objects A and B:

  • Before collision:
    $p<em>{tot,before} = m</em>A v<em>A + m</em>B v_B$

  • After collision:
    $p<em>{tot,after} = m</em>A v'<em>A + m</em>B v'_B$

  • Momentum conservation ensures
    $p<em>{tot,before} = p</em>{tot,after}$

  • Individual velocities change during the collision, but total momentum remains constant.

Types of Collisions

• Elastic Collision:

  • Definition: Total kinetic energy (KE) is conserved. (KE before = KE after)

  • Example: A rubber ball bouncing on concrete shows minimal energy loss; KE is converted to elastic potential energy (PE) upon compression and back to KE while decompressing.

• Inelastic Collision:

  • Some kinetic energy is lost to other forms of energy (thermal, sound, etc.).

• Perfectly Inelastic Collision:

  • Two objects collide and stick together post-collision.

  • Note: All macroscopic collisions are inelastic (some kinetic energy is always lost). However, atomic-level collisions can be elastic, as in the case of air molecules.

One-Dimensional Collisions

• Direction Representation:

  • Direction of velocities (v) in one dimension uses signs:

  • (+) = right

  • (-) = left

  • Example velocities:

  • v_A = +2 m/s (moving right)

  • v_B = -3 m/s (moving left)

• Caution with Notation:

  • The symbol "v" may represent speed (always positive) in some contexts but represents velocity (whose sign can be either positive or negative) in 1D collision problems.

• Collisions Example (Perfectly Inelastic):

  • Objects A (moving) and B (at rest) collide and stick together:

  • Total momentum before collision:
    $p<em>{tot,before} = m</em>A v<em>A + m</em>B(0) = m<em>A v</em>A$

  • Total momentum after collision:
    $p<em>{tot,after} = (m</em>A + m_B)v'$

  • Setting them equal gives:
    $m<em>A v</em>A = (m<em>A + m</em>B)v'$

  • Therefore:
    $v' = rac{m<em>A}{(m</em>A + m<em>B)} v</em>A$

  • Note that ( v' < vA ) since ( rac{mA}{(mA + mB)} < 1 ).

Recoil Example (Gun Firing)

• Situation: A Gun (mass M) fires a bullet (mass m) with velocity ( vb ). Calculating recoil velocity of the gun (vG ) yields:
  $p<em>{tot, before}=0 = m v</em>b + M v_G$

  • Solving gives:
    $M v<em>G = -m v</em>b$

  • Therefore:
    $v<em>G = - rac{m}{M} v</em>b$

  • Given (vb = 500 m/s), (m = 0.01 kg), and (M = 3 kg): $v</em>G = - rac{0.01 kg imes 500 m/s}{3 kg} = -0.167 m/s$

  • Interpretation: recoil occurs (gun’s backward motion when bullet is fired).

• Rocket Functionality Explanation: Rockets function based on this principle of momentum transfer (mass ejected backward causes forward propulsion).

Impulse and Momentum Conservation

• Impulse: A concept relating force to the change in momentum. Defined as:
  $J = F_{net} imes ext{time}$

  • Newton emphasized the relationship:
    $F_{net} = rac{dp}{dt}$

  • The net force is the rate of change of momentum.

• Impulse Equation with Variable Force:
  $J = ext{net} F_{avg} imes ext{d}t = rac{d}{dt}\bigg( ext{Force} imes t \bigg) = ext{net} rac{ ext{d}p}{ ext{d}t}$

• Example with Baseball:

  • Given a baseball (mass 0.30 kg), and a collision duration ( igtriangleup t = 0.01s ): impact forces yield:

  • Initial velocity (vi = -42 m/s), final (vf = 80 m/s):

  • Impulse calculation yields:

  • $J = ext{mass} imes (v<em>f - v</em>i) = 0.30 kg imes (80m/s - (-42m/s))$

  • Therefore:
    $J hickapprox 37 kg imes m/s$
    • Impulse direction is to the right (indicating net directional momentum).

  • Average force exerted during the collision is calculated as:

  • $F = rac{J}{\bigtriangleup t} hickapprox 3700 N$

Proof of Momentum Conservation in Collisions

• Proof Overview:

  • When objects A and B collide, each experiences equal and opposite forces (Newton’s Third Law).

  • Change in momentum of A equals the negative change in momentum of B:

  • $\bigtriangleup p<em>A = -\bigtriangleup p</em>B$

  • Total momentum change in isolated system:

  • $\bigtriangleup (p<em>A + p</em>B) = 0 ext{ gives } p_{tot} = constant$

  • Note: Momentum conservation holds if the system is isolated (without external forces affecting).

Ballistic Pendulum Example

• Description: Used to measure bullet speed: consists of a hanging block hitting a bullet, which embeds into it and swings upwards.

• Momentum and KE Analysis:

  • Initial Momentum:
    $m v<em>1 = (M + m)v</em>2$

  • Initial KE is partially converted to thermal energy.

• After swinging, calculate max height reached (h), using:
  $KE<em>{initial} = PE</em>{final} ext{ and } h = rac{KE}{M+m ext{g}}$

Elastic Collisions

• Eulerian References:

  • For both momentum (p) and kinetic energy (KE), conservation laws apply:

  • $m<em>A v</em>A + m<em>B v</em>B = m<em>A v'</em>A + m<em>B v'</em>B$ (momentum)

  • $rac{1}{2} m<em>A v</em>A^2 + rac{1}{2} m<em>B v</em>B^2 = rac{1}{2} m<em>A (v'</em>A)^2 + rac{1}{2} m<em>B (v'</em>B)^2$ (kinetic energy)

• Key Equation: For elastic collisions, relative speeds before and after the collision relate via:
  $(v<em>A - v</em>B) = -(v'<em>A - v'</em>B)$

• Practical Solution: Using these principles can let one solve for final velocities easily.

• Example Collision: When mass (10m) with velocity collides with resting mass (m) yields:

  • Applying conservation principles:
    $A<em>A = 10m v</em>A + 0 = 10m v<em>A' + m v</em>B'$

  • Followed with elastic collision relationship, yields:
    Final calculation gives values for velocities after the collision.

  • Summary of final velocities may be stated as:
    $v<em>A' = rac{(m</em>A - m<em>B)v</em>A + 2m<em>B v</em>{B}}{(m<em>A + m</em>B)}$
    $v<em>B' = rac{(m</em>B - m<em>A)v</em>B + 2m<em>A v</em>{A}}{(m<em>A + m</em>B)}$

Center of Mass of Extended Objects

• Definition: The point that behaves as if all mass were located at this point. Given by:
  $RM = rac{1}{M} ext{ ext{Sum}}<em>{i=1}^{N} m</em>i r_i$

  • For an extended object with many particles, it simplifies computation.

  • Location Calculation: Given a set of four masses in a square, through weighted averages define center of mass.

• Velocity and Acceleration: Defined as:

  • Velocity of c.m.:
    $V = rac{ ext{Sum}<em>{i=1}^{N} m</em>i v_i}{M}$

  • Acceleration of c.m.:
    $A = rac{ ext{Sum}<em>{i=1}^{N} m</em>i a_i}{M}$

  • Dynamics governing c.m. equivalent to point mass dynamics.

• Overall Dynamics: The total external forces acting on a system affect the motion of the center of mass, while internal forces cancel out.

• Practical Example: Upon explosion during trajectories and projectiles, c.m. behaves akin to the point mass pre-explosion.

Conclusion

• The principles of momentum conservation apply rigorously for both isolated systems and particles subject to collective interactions, with practical examples ranging from classical mechanics to real-world applications like rocket propulsion and collision analyses.