momentum

Momentum

Definition of Momentum

• Momentum combines the concepts of mass and velocity of an object.

• It is defined as the quantity of motion that an object possesses.

• Momentum can be described as a measurement of mass in motion: how much mass is in how much motion.

Stopping Momentum

• To bring an object to a complete stop, its momentum must be decreased to 0.

Calculation of Momentum

Basic Equations

• Momentum, denoted by P, is calculated using the formula: $P = mv$ Where:

  • $P$ = momentum (measured in kg•m/s or Newton-seconds, N•s)

  • $m$ = mass (measured in kg)

  • $v$ = velocity (measured in m/s)

Example Calculations

Train Momentum Calculation

1. For a 100,000 kg train moving at 5.0 m/s:

   • Calculation:
     $P = (100,000 ext{ kg})(5.0 ext{ m/s})$
     = $500,000 ext{ kg•m/s}$ or N-s.

2. For a 100 kg train moving at 5.0 m/s:

   • Calculation:
     $P = (100 ext{ kg})(5.0 ext{ m/s})$
     = $500 ext{ kg•m/s}$ or N-s.

Toy Car Momentum Calculation

1. For a 10 kg toy car moving at 5 m/s:

   • Calculation:
     $P = (10 ext{ kg})(5 ext{ m/s})$
     = $50 ext{ kg•m/s}$ or N-s.

2. For a 10 kg toy car moving at 10 m/s:

   • Calculation:
     $P = (10 ext{ kg})(10 ext{ m/s})$
     = $100 ext{ kg•m/s}$ or N-s.

Cart Momentum Calculation

1. Cart 1 (3.00 kg cart moving at 30 m/s North):

   • Calculation:
     $P = (3.00 ext{ kg})(30 ext{ m/s})$
     = $90 ext{ kg•m/s}$ or N-s.

2. Cart 2 (2.00 kg cart moving at 45.0 m/s South):

   • Calculation:
     $P = (2.00 ext{ kg})(45.0 ext{ m/s})$
     = $90 ext{ kg•m/s}$ or N-s.

   • Analysis: Both carts have the same momentum in magnitude but different directions.

Momentum Insights Recap

• The relationship between velocity and momentum:

  • If velocity increases, momentum increases at the same rate.

  • If velocity decreases, momentum decreases at the same rate.

  • If mass increases, momentum increases at the same rate.

  • If mass decreases, momentum decreases at the same rate.

  • Heavier objects moving at the same velocity will have more momentum than lighter objects.

  • Faster-moving objects at the same mass will have more momentum than slower ones.

Conservation of Momentum

Fundamental Principle

• According to Sir Isaac Newton, when measuring the momentum of colliding objects before and after an event:

  • The total momentum of any isolated system remains constant. Mathematically:
    $P<em>{ ext{total before collision}} = P</em>{ ext{total after collision}}$
    or
    $mv<em>{ ext{total before}} = mv</em>{ ext{total after}}$

Example of Collision

• In the scenario of a school bus colliding with a car:

  • The total momentum before the collision equals the total momentum after the collision, assuming no net external force acts on the system.

Types of Collisions

1. Completely Inelastic Collision:

   • When two objects collide and stick together, their combined momentum after the collision:

   • $ext{Momentum of bus (before)} + ext{Momentum of car (before)} = ext{Momentum of bus/car (after)}$

   • Equation:
     $(mv)<em>{ ext{bus}} + (mv)</em>{ ext{car}} = (mv)_{ ext{bus/car}}$

2. Elastic Collision:

   • In this type, the momentum conservation equation can be expressed as:

   • $(mv)<em>{ ext{bus1}} + (mv)</em>{ ext{car1}} = (mv)<em>{ ext{bus2}} + (mv)</em>{ ext{car2}}$

   • This highlights how momentum is distributed among separate wrecks post-collision.

Activity Understanding and Practice

Activity #1 - Momentum Questions

1. Calculate the final momentum when a 4000 kg bus travelling at 80 km/h collides with a car weighing 1000 kg moving at 60 km/h.

2. If a bicycle has a momentum of 24 kg•m/s, determine the new momentum when:

   • it has three times the mass and is moving at half the speed.

Impulse

Definition of Impulse

• Impulse is defined as the force acting on an object multiplied by the time interval that the force is exerted.

• It is a vector quantity, represented mathematically as: $I = F imes t$ Where:

  • $I$ = Impulse (measured in N•s)

  • $F$ = Force (measured in N)

  • $t$ = Time (measured in s)

Application of Impulse

• To reduce an object's momentum to zero, an impulse must be applied.

• Impulse influences momentum change; increasing force or time results in greater impulse.

  • Two strategies for increasing impulse include:

  1. Increase the force applied.

  2. Increase the duration the force is applied.

Observations from Practical Examples

Example: Using a Bedsheet for an Egg Drop

• Using a bedsheet to catch an egg allows the force of the egg to distribute over a wider area and a longer time:

  • This reduces the average force exerted on the egg.

  • Similar concepts apply to vehicle safety designs, such as airbags, which extend the time of impact and reduce force, enhancing safety.

Impulse in Sports

Improving Sports Performance

Volleyball Spike Example

• To improve your spike in volleyball, the impulse can be increased by:

  1. Strength training (building up strength increases the force applied).

  2. Following through on the hit (this increases contact time with the ball).

Impulse Versus Momentum Relationship

• Impulse is also defined as change in momentum:

  • $I = ext{Change in momentum} = \Delta P = m \Delta v = m(v<em>2 - v</em>1)$