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{tot,before} = mA vA + mB v_B

    • After collision:
      p{tot,after} = mA v'A + mB v'_B

    • Momentum conservation ensures
      p{tot,before} = p{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{tot,before} = mA vA + mB(0) = mA vA

    • Total momentum after collision:
      p{tot,after} = (mA + m_B)v'

    • Setting them equal gives:
      mA vA = (mA + mB)v'

    • Therefore:
      v' = rac{mA}{(mA + mB)} vA

    • 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{tot, before}=0 = m vb + M v_G

    • Solving gives:
      M vG = -m vb

    • Therefore:
      vG = - rac{m}{M} vb

    • Given (vb = 500 m/s), (m = 0.01 kg), and (M = 3 kg): vG = - 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}igg( ext{Force} imes t igg) = 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 (vf - vi) = 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}{igtriangleup 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:

    • igtriangleup pA = -igtriangleup pB

    • Total momentum change in isolated system:

    • igtriangleup (pA + pB) = 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 v1 = (M + m)v2

    • Initial KE is partially converted to thermal energy.

  • After swinging, calculate max height reached (h), using:
    KE{initial} = PE{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:

    • mA vA + mB vB = mA v'A + mB v'B (momentum)

    • rac{1}{2} mA vA^2 + rac{1}{2} mB vB^2 = rac{1}{2} mA (v'A)^2 + rac{1}{2} mB (v'B)^2 (kinetic energy)

  • Key Equation: For elastic collisions, relative speeds before and after the collision relate via:
    (vA - vB) = -(v'A - v'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:
      AA = 10m vA + 0 = 10m vA' + m vB'

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

    • Summary of final velocities may be stated as:
      vA' = rac{(mA - mB)vA + 2mB v{B}}{(mA + mB)}
      vB' = rac{(mB - mA)vB + 2mA v{A}}{(mA + mB)}

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}}{i=1}^{N} mi 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}{i=1}^{N} mi v_i}{M}

    • Acceleration of c.m.:
      A = rac{ ext{Sum}{i=1}^{N} mi 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.