Momentum, Impulse, and Energy

Chapter 6: Momentum

6.1 Momentum

• Definition of Momentum: Momentum is defined as the product of an object's mass and its velocity. It is often referred to as inertia in motion.

  • Formula:
    ext{Momentum} = ext{mass} imes ext{velocity}

  • When direction is not important, we can simplify it to speed:
    ext{Momentum} = ext{mass} imes ext{speed}

  • The symbol p is used to denote momentum:
    p = mv

• Examples of Momentum:

  • A heavy truck moving at the same speed as a small car has greater momentum due to its larger mass.

  • A small bullet moving quickly can have more momentum than a large ship moving slowly.

  • A still truck has zero momentum since velocity (v) is zero.

Check Point 1

1. When can a 1000-kg car and a 2000-kg truck have the same momentum?

   • The car will have the same momentum as the truck when it travels twice as fast:
     1000 ext{ kg} imes 2v = 2000 ext{ kg} imes v

   • Both have zero momentum when at rest.

2. How does the concept of inertia relate to momentum?

   • Momentum can be considered as “inertia in motion”.

3. A Mars rover in a trial run moves at 12 km/h on Earth and on Mars. Where is its momentum greater?

   • The rover's momentum is the same in both locations since its mass remains unchanged and speed is identical.

6.2 Impulse

• Definition of Impulse: The quantity of force multiplied by the time interval over which it acts.

  • Formula:
    ext{Impulse} = F imes t

• Effect of Impulse:

  • If an object's momentum changes, either its mass or velocity or both must change.

  • For constant mass, a force results in acceleration and hence a change in momentum.

  • The time duration during which a force acts is critical; a longer duration yields a greater change in momentum.

Check Point 1

1. Does a moving object have impulse?

   • No, impulse is experienced when interacting with another object and is not something an object possesses.

2. Does a moving object have momentum?

   • Yes, a moving object possesses momentum relative to a frame of reference, such as the Earth's surface.

6.3 Impulse-Momentum Relationship

• The impulse-momentum relationship states that impulse is equal to the change in momentum:
  ext{Impulse} = ext{change in momentum} = Ft = riangle (mv)

Case Scenarios

1. Increasing Momentum:

   • To increase momentum, apply the greatest force for a long duration.

   • Examples: A golfer or baseball player follows through with their swings, extending the time of contact and utilizing average force over varying periods.

Check Point 1

1. Which cannon imparts greater speed to a cannonball: a long-barreled or a short-barreled cannon?

   • A long-barreled cannon imparts greater speed due to applied force over a longer time, resulting in more impulse.

2. In blowing peas from straws of different lengths, which straw will propel the pea further?

   • A pea shot from a longer straw will travel farther because it experiences greater speed, adhering to the same physics as the cannon example.

• Vector Quantities:

  • Both impulse and momentum are vector quantities; direction is as significant as magnitude.

Case 2: Decreasing Momentum Over a Long Time

• In an out-of-control car, hitting a haystack extends stopping time compared to hitting a concrete wall, reducing force and deceleration:

  • Impulse remains constant, but no force applied long means lesser force/impact.

• Examples: Padded dashboards and airbags save lives by increasing stopping time in collisions.

Case 3: Decreasing Momentum Over a Short Time

• Situations with short contact times leading to large contact forces:

  • Boxing: Moving into a punch minimizes the time of impact, increasing force.

  • Catching a baseball with an extended hand reduces contact force.

6.4 Bouncing

• Bouncing Impulse: Impulses are greater when an object bounces.

  • Example: When catching a falling flowerpot, catching it reduces momentum to zero; if it bounces, more impulse is required to stop and throw it back up.

• Related example: The California Gold Rush employed bouncing water in paddle wheels for efficiency.

Check Point 1

1. How does the force Cassy exerts on bricks compare to the force on her hand?

   • According to Newton's third law, forces exerted on both sides are equal.

2. If Cassy's hand bounces back, how is impulse affected?

   • Greater impulse develops if the hand bounces, thus increasing the force on both bricks and her hand.

6.5 Conservation of Momentum

• Conservation Principle: No external impulse can change a system's momentum.

  • Only external forces or impulses can change momentum; internal forces do not affect it.

• Systems: Internal forces cancel out in a closed system; external forces must act to change momentum.

• Examples in Pool:

  • 8-Ball System: External force gained by the 8-ball alters its momentum when hit by a cue ball.

  • Cue-Ball System: Reaction forces changing momentum reflect external impacts when the 8-ball strikes.

  • Combined System:

  • Total momentum before and after collisions remains the same demonstrating internal motions.

• Elastic vs Inelastic Collisions:

  • Elastic: Colliding objects rebound forming no lasting deformation or heat.

  • Inelastic: Objects become distorted, generate heat, and potentially stick together, but momentum is always conserved.

Check Point 1

• If a ball is tossed horizontally while on a skateboard, will it roll backward?

  • No external impulse means no backward force. The skateboard will not roll backward.

• Cannon Function: Recoil and fired ball create opposite forces internal to the cannonball-cannon system; total momentum remains zero.

• Conservation of Momentum: A system's momentum persists when no external forces act on it.

6.6 Collisions

• The net momentum of colliding objects remains unchanged; forces acting in collisions are internal.

  • During an elastic collision, a moving billiard ball striking another at rest transfers momentum perfectly.

• Inelastic collision example: Two freight cars crash, couple, and move together maintain their momentum distribution.

Example Calculations

• If the first freight car of mass m moves at 10 m/s, its momentum before collision is calculated as mv = 10m

  • If two equal massive freight cars collide and couple become 2m, their velocity becomes 5 m/s post-collision ensuring momentum conservation:
    mv + 0 = (m + m)v'

Scenarios of Colliding Objects

1. Equal Momentum in Opposite Directions:

   • Total momentum equals zero; neither ball moves away in the collision.

2. Simultaneously Moving Objects:

   • When cars move in the same direction they add to total momentum.

6.7 More Complicated Collisions

• Net momentum remains unchanged in any collision and is calculated using vector addition laws.

  • Example: Two cars collide at right angles, resulting momentum's vector is calculated using the parallelogram law.

  • Resultant momentum is not simply the arithmetic sum; it relies on vector magnitudes.

Summary Situations

1. When riding twice as fast on a bicycle, momentum is also doubled (direct relationship with velocity).

2. An iron ball and a wooden ball dropped simultaneously will hit the ground at the same speed, but the iron ball has greater momentum.

3. A long-barrel cannon will give cannonballs more impulse due to extended time for force application.

4. A freight train at the same speed but double mass has doubled momentum.

5. Collision momentum is conserved in both elastic and inelastic cases.

Small Problems

1. A rolling freight car collides with a still one; if together they speed up to 2 m/s, the initial speed must have been 4 m/s.

2. The more massive the bus, the greater the braking force needed to stop it in limited distance.

3. Impulse varies based on stopping conditions; hitting a haystack versus a wall reflects this trade-off.

7. Energy

7.1 Work

• Definition of Work: Work is done when a force acts on an object to move it a distance: W = ext{force} imes ext{distance}

  • Work requires both application of force and movement.

• Conditions for Work:

  • Work is zero if the object does not move, irrespective of how much force is applied.

• Units:

  • Measured in joules (J), where 1 J = 1 N·m.

Check Point 1

1. Work lifting a bag of groceries weighing 200 N to a height of 3 m:
   W = 200N imes 3m = 600J

2. Lifting the bag twice as high:

   • Requires twice the work, leading to 1200 J.

7.2 Power

• Definition of Power: Power is the rate of doing work:
  ext{Power} = rac{ ext{work}}{ ext{time}}

• Unit of Power:

  • Joule per second (J/s), or watts (W).

  • 1 horsepower = 746 W; engines rated on this standard.

Check Point Calculations

• Lifting a 4000 N piano distance of 4 m in 2 seconds implies 8000 W of power.

• To accomplish this in 1 second requires double the power (16,000 W).

7.3 Potential Energy

• Definition of Potential Energy: Stored energy based on position (e.g., raised objects).

  • Relevant Formula:
    ext{Potential Energy} = mgh

  • Meaning work done against gravity when moved is stored energy used.

Check Point Energy Calculations

1. Increase in gravitational potential energy is consistent regardless of path taken to lift.

2. Examples include energy stored in springs, chemicals in fuel, and elevated water.

7.4 Kinetic Energy

• Kinetic Energy Definition: Energy of motion defined by the formula:
  KE = rac{1}{2} mv^2

• Work is done to give any object its kinetic energy.

Check Points:

1. Example: A ball’s throw gives it speed, and it can do work after being set in motion.

7.5 Work-Energy Theorem

• Law Formulation: Work done equals the change in kinetic energy due to external forces acting against it.
  ext{Work} = riangle KE

Setting Up Equations

• Work done by friction or net forces on objects reflects in overall kinetic energy.

7.6 Conservation of Energy

• Law of Conservation: Energy cannot be created or destroyed; it merely transforms, maintaining total amount steady.

• Transformation includes cases where potential energy converts into kinetic energy or vice versa.
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7.7 Machines

• Machines multiply force or change direction of force.

• Mechanical Advantage: More input leads to more output.

Example of Losses

1. With any machine, no energy can be multiplied; it can only be transformed or transferred.

2. Efficiency affects output compared to input work and highlights thermal loss.

7.8 Efficiency

• Efficiency is expressed as:

  • ext{Efficiency} = rac{ ext{useful work output}}{ ext{total work input}}

• Inefficiencies lead to thermal energy loss.