Impulse & Momentum

AP PHYSICS 1 NOTES: IMPULSE, MOMENTUM, AND IMPULSE-MOMENTUM THEOREM

I. OVERVIEW OF MOMENTUM

• Definition of Momentum:

  • The momentum $p$ of a particle is a vector quantity defined as:
    $p = mv$

  • Where:

  • $m$ = mass of the object (in kg)

  • $v$ = velocity of the object (in m/s)

  • Characteristics of momentum:

  • Momentum quantifies an object's motion, with direction aligned with the object's velocity.

  • Two types of momentum:

    • Negative momentum: Objects moving backward (indicated with a negative sign).

    • Positive momentum: Objects moving forward (indicated with a positive sign).

Sample Question 1: Change in Momentum

• Question: What is the cart's change in momentum?

• Before Momentum Calculation:

  • Initial Velocity $V_1 = -2 ext{ m/s}$

  • Final Velocity $V_2 = +1 ext{ m/s}$

  • Change in Momentum:
    $DP = m(V<em>2 - V</em>1) = m(1 - (-2)) = m(3)$

Sample Question 2: Change in Magnitude of Momentum

• Question: In which case is the change in the magnitude of the momentum greatest?

  • Factors include:

  • Mass of the object

  • Change in velocity ( d_v )

  • Change in direction

  • Example Calculation:
    $ext{Change in Momentum} = m(v<em>2 - v</em>1)$(for a 10 kg mass).

II. FORCE AND MOMENTUM

• Definition:

  • Momentum describes how hard it is to stop or turn a moving object—the application of a force is necessary to overcome momentum.

• Changing Momentum:

  • Changes in momentum are caused by forces, and applying a force changes an object's momentum.

  • Key points:

  • The more mass an object has, the more force it takes to change its momentum.

  • The faster an object is moving, the more force it takes to change its momentum.

Forces: Action-Reaction

• Newton's Third Law:

  • Forces always come in pairs when two objects collide; they exert equal and opposite forces on each other.

  • Example Scenario:

  • Object A exerts Force on Object B, while Object B exerts an equal and opposite force on Object A.

Sample Problem 3: Momentum Impedance

• Scenario: A 15-g bullet moving at 300 m/s passes through a 2.0 cm thick sheet of foam plastic and emerges with a speed of 90 m/s.

• Calculation of average force impeded during motion:

  • Initial velocity $V_0 = 300 ext{ m/s}$

  • Final velocity $V_f = 90 ext{ m/s}$

  • Distance $D_X = 0.02 ext{ m}$

  • Average Force $F_{avg}$ is obtained through the change in momentum.

III. IMPULSE

• Definition of Impulse (J):

  • Impulse is defined as the change in momentum. Symbols for impulse:

  • $J = riangle p$

  • Units of Measurement:

  • Impulse: N·s

  • Momentum: kg·m/s

• Relationship between Force and Time:

  • The change in momentum occurs only when a force acts over a duration.

  • The shorter the time, the more force is required to produce the same change in momentum:

  • Car Example:

  • Stopping a 1,000 kg car from 30 m/s using a 3,000 N force over 10 seconds demonstrates impulse-related calculations.

Collision Duration

• Definition: A collision is a short-term interaction wherein materials' properties impact the duration of contact.

  • Increasing contact time decreases the force of impact, illustrated by air bags in cars.

  • Compression of materials, such as a soccer ball, indicates applied force magnitude.

Impulse Representation

• Equation: $J = F riangle t = m(v - v_1)$

  • Where $F$ is the average force acting over time $t$.

Characteristics of Impulsive Forces

• Impulsive forces are generally characterized as high magnitude and short duration, indicating large forces when momentum changes quickly.

IV. IMPULSE-MOMENTUM THEOREM

• Theorem Statement:

  • The impulse experienced by an object equals the change in momentum:
    $J = riangle p$

• Newton’s 2nd Law Connection:

  • The Impulse-Momentum Theorem expresses the same principles as Newton's Second Law:

  • $F riangle t = m riangle v = J$

Sample Problem 4: Brick Impact

• Scenario: A brick of 1.5 kg is dropped from 20 cm height.

  • Calculate impulse necessary to stop the brick:

  • Initial Velocity at impact can be derived through energy conversion (using gravitational potential energy).

• Results indicate springboard calculations for impulse when halting the brick in the given time frame.

Sample Problem 5: Sledgehammer Impact

• Scenario: A 3 kg sledgehammer traveling at 14 m/s stops in 0.02 seconds. Find the average force exerted on the spike during the event.

V. MOMENTUM AT ANGLES

• Momentum as a Vector: Momentum is treated as a vector, necessitating consideration of directions when adding momenta;

• Resultant Momentum Calculation: $P<em>{total} = P</em>1 + P<em>2 + P</em>3$

  • Where each momentum vector is combined.

VI. FORCE VS. TIME GRAPHS

• Understanding Area Under the Curve:

  • The area under a force vs. time graph corresponds to impulse (measured in kg·m/s).

  • Impulse (J) evaluates as the integral of force over time.

• Impulses can fluctuate in magnitude throughout the interval, often starting high and tapering off.

Sample Problem 10: Force Impact on Velocity

• Scenario: A 1.2 kg object moving at 120 m/s has a force applied, changing its velocity. Assessment of new velocity post-force application.

Homework Exercises: Impulse-Momentum Theorem

1. Problem involving a car's mass, momentum, and braking time.

2. Seek bullet change in momentum and average force at discharge.

3. Boy's skateboard force analysis for push-off.

4. Determine changes in momentum and average force of an arrow upon discharge.