Linear Momentum and Collisions: Study Notes (Chapter 9)

Linear Momentum and Collisions

9-1 Linear Momentum

Linear momentum is a fundamental concept in physics that describes the quantity of motion an object possesses. It is a vector quantity, meaning it has both magnitude and direction.

• Definition: The linear momentum p of an object is defined as the product of its mass m and velocity v: $\vec{p} = m\vec{v}$

• SI Unit: kg·m/s

• Direction: The direction of momentum is the same as the direction of velocity.

• Change in Momentum: The change in momentum depends on the initial and final velocities. For example, if an object reverses direction, the change in momentum is $2mv$.

• Example:

  • (a) What is the magnitude of the momentum of a 0.0053-kg marble whose speed is 0.65 m/s? $p = (0.0053\,\text{kg})(0.65\,\text{m/s}) = 0.003445\,\text{kg·m/s}$

  • (b) What is the speed of a 0.132-kg baseball whose momentum has a magnitude of 3.28 kg·m/s? $v = \frac{p}{m} = \frac{3.28}{0.132} = 24.85\,\text{m/s}$

9-2 Momentum and Newton’s Second Law

Newton’s Second Law relates force to the rate of change of momentum. The general form is valid even when mass changes.

• Standard Form: $\sum \vec{F} = m\vec{a}$ (valid for constant mass)

• General Form: $\sum \vec{F} = \frac{d\vec{p}}{dt}$

• Application: Useful for systems where mass is not constant, such as rockets.

9-3 Impulse

Impulse quantifies the effect of a force acting over a time interval, resulting in a change in momentum.

• Definition: $\vec{I} = \vec{F}_{\text{av}} \Delta t$

• SI Unit: N·s = kg·m/s

• Impulse-Momentum Theorem: $\vec{I} = \Delta \vec{p}$

• Interpretation: The same change in momentum can be produced by a large force acting briefly or a small force acting longer.

• Example:

  • A baseball (m = 0.14 kg) approaches a bat at $v_i = -38$ m/s and leaves at $v_f = +58$ m/s. (a) Impulse: $I = m(v_f - v_i) = 0.14 \times (58 - (-38)) = 0.14 \times 96 = 13.44$ kg·m/s (b) Average force if contact time is 1.6 ms: $F_{\text{av}} = \frac{I}{\Delta t} = \frac{13.44}{0.0016} = 8400$ N

9-4 Conservation of Linear Momentum

Momentum is conserved in a system if the net external force is zero. Internal forces do not affect the net momentum of the system.

• Conservation Law: If $\sum \vec{F}_{\text{ext}} = 0$, then $\vec{p}_i = \vec{p}_f$

• Internal vs. External Forces:

  • Internal forces act within the system and always sum to zero.

  • External forces are responsible for changes in the system’s momentum.

• System of Objects: $\vec{p}_{1,i} + \vec{p}_{2,i} + \cdots = \vec{p}_{1,f} + \vec{p}_{2,f} + \cdots$

• Example:

  • A small car and a large truck collide and stick together. Both experience the same magnitude of momentum change, but in opposite directions.

  • Amy (150 lbs) and Gwen (50 lbs) push off each other on ice. If Amy slides at 6 m/s, Gwen’s speed is determined by conservation of momentum: $m_{Amy}v_{Amy} = m_{Gwen}v_{Gwen}$, so $v_{Gwen} = \frac{150}{50} \times 6 = 18$ m/s.

  • A rifle (3.0 kg) fires a 5.0 g bullet at 1200 m/s. The recoil speed of the rifle: $v_{rifle} = -\frac{m_{bullet}v_{bullet}}{m_{rifle}}$

9-5 Inelastic Collisions

In inelastic collisions, momentum is conserved but kinetic energy is not. In a completely inelastic collision, objects stick together after impact.

• Definition:

  • Inelastic collision: $\text{momentum conserved, kinetic energy not conserved}$

  • Completely inelastic: objects stick together

• Final velocity after collision: $v_f = \frac{m_1v_{1i} + m_2v_{2i}}{m_1 + m_2}$

• Example:

  • A 1180 kg car (+24 m/s) collides with a 2470 kg truck (-16 m/s), stick together. $v_f = \frac{1180 \times 24 + 2470 \times (-16)}{1180 + 2470}$

• Ballistic Pendulum:

  • Used to measure projectile speed. Conservation of momentum applies during collision; conservation of energy applies after.

9-6 Elastic Collisions

Elastic collisions conserve both momentum and kinetic energy. They can be analyzed in one or two dimensions.

• One-Dimensional Elastic Collision:

  • Final velocities: $v_{1f} = \frac{m_1 - m_2}{m_1 + m_2}v_{10}$ $v_{2f} = \frac{2m_1}{m_1 + m_2}v_{10}$

• Two-Dimensional Collisions:

  • Require knowledge of some final velocities or angles to solve.

• Example:

  • Two pucks collide on an air hockey table; use conservation laws to find final speeds.

9-7 Center of Mass

The center of mass is the point where the mass of a system is considered to be concentrated for analysis of motion in a gravitational field.

• Definition: The center of mass (CM) is the balance point of a system.

• For Two Objects: $x_{cm} = \frac{m_1x_1 + m_2x_2}{m_1 + m_2}$

• Generalization for Many Objects: $x_{CM} = \frac{\sum m_ix_i}{M}$ $y_{CM} = \frac{\sum m_iy_i}{M}$

• Properties:

  • The center of mass is closer to the more massive object.

  • It need not be within the object itself.

  • For symmetric, uniform objects, the CM is at the geometric center.

• Motion of the Center of Mass:

  • Velocity: $\vec{V}_{cm} = \frac{\sum m_i\vec{v}_i}{M}$

  • Acceleration: $\vec{A}_{cm} = \frac{\sum m_i\vec{a}_i}{M}$

  • Newton’s Second Law for a System: $M\vec{A}_{cm} = \vec{F}_{net,ext}$

• Example:

  • Three particles at the vertices of an equilateral triangle: use the above formulas to find the CM coordinates.

Additional info:

• Questions and examples throughout the notes reinforce conceptual understanding and problem-solving skills.

• Tables and diagrams (not shown here) illustrate collision types, conservation laws, and center of mass calculations.