Lecture 6: Momentum and Collisions

Introduction to Momentum: Inertia in Motion

• Momentum is defined as inertia in motion. It is a fundamental property used to quantify the motion of an object.

• Understanding momentum, alongside energy (to be discussed in future lectures), provides essential insights into the mechanics of the universe.

• Momentum allows for the analysis of object motion both in isolation and during interactions, such as collisions between multiple bodies.

Definition and Fundamental Properties of Momentum

• Quantitative Definition: Momentum is a property that characterizes the motion of an object based on its mass and velocity.

• Vector Nature: Momentum is a vector quantity, meaning it possesses both a magnitude and a specific direction. The direction of the momentum vector is always the same as the direction of the velocity vector.

• Symbol: The lowercase letter $p$ represents momentum.

• Units: The standard units for momentum are kilogram-meters per second ($kg \cdot m/s$) or Newton-seconds ($N \cdot s$). These units are interchangeable and equivalent.

Determining Momentum: Proportionality of Mass and Velocity

• Momentum is derived from the combination of mass (the source of inertia) and velocity (the definition of the object's motion).

• Direct Proportionality to Mass: Momentum increases linearly with an object's mass when velocity is constant.

  • In an experimental trial using a cart released from a constant height (constant speed) onto a force probe:

    • A mass of $340\,g$ resulted in a recorded impact force of $29.4\,N$.

    • Increasing the mass to $440\,g$ increased the force to $32.1\,N$.

    • Increasing the mass further to $540\,g$ increased the force to $34.4\,N$.

• Direct Proportionality to Velocity: Momentum increases linearly with an object's speed when mass is constant.

  • In an experimental trial keeping the cart mass at $540\,g$:

    • At a lower release height (lower speed), the force probe recorded $34.4\,N$.

    • At a higher release height (higher speed), the force probe recorded $45.4\,N$.

• Vector Consistency: In a collision where a moving red cart impacts a stationary blue cart on a low-friction track, the momentum transfers to the blue cart, cause it to move in the same direction as the initial motion of the red cart.

Calculating Momentum: Scalars, Vectors, and Directions

• The Momentum Equation: The formula for calculating momentum is:     $p = m \times v$

• Conversion Requirements: All mass values must be converted to kilograms ($kg$) and velocity to meters per second ($m/s$).

• Sample Calculation: For a cart with a mass of $0.34\,kg$ and a measured velocity of $0.42\,m/s$ at the moment of impact:     $p = 0.34\,kg \times 0.42\,m/s = 0.143\,kg \cdot m/s$

• The Importance of Direction: Two objects with identical masses and speeds do not have equal momentums if they are traveling in different directions, due to the vector nature of the property.

Impulse: Quantifying the Change in Momentum

• Definition: Impulse is the measure of the change in momentum of a single object or a system of objects.

• Symbol: The symbol for impulse is $\Delta p$, utilizing the Greek letter delta to represent change.

• Equation 1 (Definition): Impulse is calculated as the difference between the final and initial momentum:     $\Delta p = p_f - p_i$

• Vector Nature: Like momentum, impulse is a vector quantity. Careful attention must be paid to the positive or negative signs of the variables during calculations to reflect direction.

The Relationship between Impulse, Force, and Interaction Time

• Changes in momentum are achieved by applying a force over a span of time.

• Impulse-Force Proportionality: Higher forces applied over a constant time interval result in a greater change in momentum.

  • Experiment: Using a hanging mass and pulley to pull a cart, increasing the hanging mass (increasing force) led to a larger change in the cart's momentum recorded by a motion detector.

• Impulse-Time Proportionality: A constant force applied over a longer duration results in a greater change in momentum.

  • Experiment: Applying a fixed force to a cart over increasing distances/durations showed that longer contact times produced higher final momentum.

• Equation 2 (Impulse-Momentum Theorem): Impulse is the product of force and the time interval during which that force acts:     $\Delta p = F \Delta t$

• Maximizing Momentum Change: To achieve the largest possible change in momentum, a large force must be exerted over a long period of time.

• Directional Consistency: The force vector and the impulse vector point in the same direction.

  • To speed an object up: Apply force in the same direction as velocity.

  • To slow an object down: Apply force in the opposite direction of velocity.

Impact Dynamics: Comparing Bouncing and Non-Bouncing Collisions

• Bouncing involves a greater change in momentum (impulse) than merely stopping.

• Experimental Demonstration: Two nearly identical balls (mass $0.01\,kg$) are dropped onto a block of wood.

  • The non-bouncing ball deflects the wood slightly. The wood must only apply enough force to stop the ball's downward motion.

  • The bouncing ball knocks the wood over. The wood must apply enough force to stop the ball AND provide the additional impulse required to move it in the opposite direction.

• Quantitative Comparison:

  • Initial Conditions for both: $m = 0.01\,kg$, $v_i = -2.5\,m/s$ ($p_i = -0.025\,kg \cdot m/s$).

  • Non-bouncing Ball: $v_f = 0$, $\Delta p = 0 - (-0.025) = +0.025\,kg \cdot m/s$.

  • Bouncing Ball: Rebounds at $v_f = +1.5\,m/s$ ($p_f = +0.015\,kg \cdot m/s$). $\Delta p = 0.015 - (-0.025) = +0.04\,kg \cdot m/s$.

• Conclusion: The impulse is significantly higher ($0.04$ versus $0.025$) when an object bounces.

The Law of Conservation of Momentum

• Core Principle: In the absence of external forces, the total momentum of a system is conserved (remains constant).

• Experimental Evidence: A cart pushed on a low-friction track maintains nearly constant speed. However, if a high-friction pad is added, friction acts as an external force that rapidly decreases momentum, demonstrating that conservation only holds without significant external influence.

Systems and External Forces

• The conservation of momentum depends on how the "system" is defined.

• Dual-Cart System:

  • Blue cart ($m = 0.34\,kg$) hits a stationary Red cart ($m = 0.34\,kg$) and they stick via Velcro.

  • Initial System Momentum: $0.083\,kg \cdot m/s$ (Blue) + $0$ (Red) = $0.083\,kg \cdot m/s$.

  • Final System Momentum: Combined mass ($0.68\,kg$) at $0.117\,m/s = 0.080\,kg \cdot m/s$.

  • Result: Closely matched values demonstrate conservation (slight loss due to minor friction).

• Single-Cart System Analysis:

  • If the system is only the Blue cart: Momentum drops from $0.083$ to $0.04\,kg \cdot m/s$. Momentum is NOT conserved because the Red cart is an "external object" providing an external force.

  • If the system is only the Red cart: Momentum increases from $0$ to $0.04\,kg \cdot m/s$. Momentum is NOT conserved because the Blue cart provides an external force.

Momentum Conservation in Multiple Dimensions

• Momentum is conserved in one, two, and three dimensions in the absence of external forces.

• Two-Dimensional (2D) Conservation: Horizontal ($x$) and vertical ($y$) components of momentum are considered independently. The sum of momentum vectors before a collision equals the sum after.

  • Example: Magnetic pucks on a low-friction air cushion table rebound without touching. The x-components and y-components are conserved separately.

• Three-Dimensional (3D) Conservation: Momentum is conserved across width, depth, and height components simultaneously.

Kinetic Energy and Work-Energy Relationships in Dynamics

• Kinetic Energy (KE): The energy an object possesses due to its motion.

• Symbol and Units: Symbol is $KE$, measured in Joules ($J$).

• Formula:     $KE = \frac{1}{2} m v^2$

• Context: Unlike momentum, kinetic energy is not always conserved in collisions; it can be transformed into heat, sound, or deformation energy.

Characterizing Collisions: Elastic and Inelastic

• Inelastic Collisions:

  • Momentum is conserved.

  • Kinetic energy is NOT conserved; it is converted into heat, sound (acoustic energy), or structural deformation.

  • Perfectly Inelastic Collision: Two objects collide and stick together, traveling as a single unit afterward. This represents the maximum loss of kinetic energy.

• Elastic Collisions:

  • Both momentum and kinetic energy are conserved.

  • Occurs when there is no heat generated from friction, no sound produced, and no physical deformation of molecules.

  • Usually involves non-contact forces like repelling magnets.

Perfectly Inelastic Collision: Case Study and Data

• Setup: Blue cart ($0.54\,kg$) hits Red cart ($0.34\,kg$) and they stick.

• Momentum Data:

  • Initial: Red ($p = 0$) + Blue ($p = 0.197$) = $0.197\,kg \cdot m/s$.

  • Final: Combined mass ($0.88\,kg$) at $0.218\,m/s = 0.192\,kg \cdot m/s$.

  • Conclusion: Momentum is conserved.

• Kinetic Energy Data:

  • Initial: $0.036\,J$.

  • Final: $0.021\,J$.

  • Conclusion: KE is not conserved (lost to heat and sound).

Elastic Collision: Case Study and Data

• Setup: Red cart ($0.54\,kg$) and Blue cart ($0.34\,kg$) with repelling magnets collide without touching.

• Momentum Data:

  • Initial: $0.099\,kg \cdot m/s$.

  • Final: $0.101\,kg \cdot m/s$.

  • Conclusion: Momentum is conserved.

• Kinetic Energy Data:

  • Initial: $0.014\,J$.

  • Final: $0.013\,J$.

  • Conclusion: KE is conserved.

Questions & Discussion: Interview with Dr. Bennett

• Background: Dr. Bennett holds a PhD in high-energy physics and worked on the Compact Muon Solenoid (CMS) at the Large Hadron Collider (LHC).

• What is the LHC? A 27-kilometer-long ring several stories underground in Switzerland and France. It functions as a "proton crasher," slamming bunches of protons together to recreate exotic, high-mass particles.

• How are conservation laws used to see the unseen? Because the exotic particles produced (like Higgs bosons or tau leptons) decay almost instantly, physicists cannot observe them directly. Instead, they measure the energy and momentum deposited in the layers of the cylindrical detector.

  • Energy Conservation: Summing the energy bits deposited in all layers allows physicists to calculate the mass of the original particle at the core of the collision.

  • Momentum Conservation: Initially, two proton beams head toward each other with equal and opposite momentum, making the total system momentum zero. After the crash, the sum of all momentum vectors of all resulting particles must still equal zero. If they do not, it indicates a particle was missed or unaccounted for.

• Difference between Elastic and Inelastic in Colliders:

  • Elastic: Protons bounce off each other or have glancing blows with no structural change. Dr. Bennett describes this as "dull" and something that needs to be filtered out of data.

  • Inelastic: This is where internal quarks recombine and energy transforms into matter. These collisions create the interesting physics, such as the production of the Higgs boson.

Engineering Safer Collisions: The Impulse-Momentum Relationship

• The goal in safety engineering (cars, bikes) is to minimize the impact force ($F$) during a collision.

• The Constraint: In a crash, the initial and final momentums are usually fixed (you start at a speed and end at zero), meaning the impulse ($\Delta p$) is a fixed value.

• Mathematical Logic: Since $\Delta p = F \Delta t$, if $\Delta p$ is constant, increasing the duration of the collision ($\Delta t$) will mathematically force the impact force ($F$) to decrease.

• Practical Examples:

  • Egg Toss: Throwing an egg at a wall results in a very small $\Delta t$, high force, and the egg breaks. Throwing an egg into a sagging sheet extends the $\Delta t$ significantly, reducing the force enough to keep the shell intact.

  • Automotive Engineering: Crumple zones and airbags are designed specifically to extend the time of the collision, thereby reducing the force exerted on the vehicle's occupants.

Quantitative Case Study: Extending Duration to Reduce Peak Impact Force

• Experiment using a cart rolling down a ramp to hit a force probe with fixed momentum.

• Scenario A: Rigid Collision (No Foam)

  • Collision Duration: approximately $0.02\,s$.

  • Maximum Force: $48.4\,N$.

  • Impulse (via Logger Pro): $0.4\,N \cdot s$.

• Scenario B: Extended Collision (With Foam)

  • Collision Duration: approximately $0.1\,s$.

  • Maximum Force: $20.6\,N$.

  • Impulse (via Logger Pro): $0.4\,N \cdot s$.

• Analysis: Even though both scenarios had the identical impulse ($0.4\,N \cdot s$), the foam scenario is safer because the maximum force was reduced by more than half ($48.4\,N$ to $20.6\,N$) by increasing the contact time five-fold.