KNES 361 L8

Lecture 8: Impulse and Linear Momentum Part Two

Overview

• The lecture discusses impulse and momentum in real-life scenarios and introduces the law of conservation of linear momentum.

Impulse

Definition

• Impulse is defined as the product of the force exerted during a specific time interval.

• Formula for Impulse:
  $\text{Impulse} = \text{Force} \times \text{Time}$.

Force vs Time Graph

• If a graph is plotted with force on the y-axis and time on the x-axis, the area under the curve represents the total impulse.

• In real-life movements, forces are often not constant — they may rise and fall, which can still be represented by the area under the force vs time curve.

Impulse Variation

• To increase impulse, one can do any of the following:

  • Increase the force applied.

  • Increase the duration during which the force is applied.

  • Do both.

• To decrease impulse, one can:

  • Decrease the force applied.

  • Decrease the duration of time that the force is applied.

  • Do both.

Example

• Applying 100 Newtons of force for 2 seconds:

  • Impulse = 100 N * 2 s = 200 Newton-seconds.

• Increasing the force to 200 Newtons while keeping time at 2 seconds:

  • Impulse = 200 N * 2 s = 400 Newton-seconds.

• Keeping the force at 100 Newtons and increasing time to 4 seconds:

  • Impulse = 100 N * 4 s = 400 Newton-seconds.

• This emphasizes that the method of increasing impulse can affect the outcome in real-life scenarios.

Importance of Impulse Duration

• If the required impulse is the same, a longer application time allows for smaller average forces which results in less peak force and potential tissue damage.

Example of Falling from a Roof

• If a person falls directly onto a hard surface (like the ground):

  • The stopping time is very short, leading to a high peak force and severe impact.

• If the same person falls onto a trampoline:

  • The trampoline stretches, increasing the time it takes to stop, resulting in smaller average force and a less severe impact.

• This principle applies to many safety devices.

Safety Mechanisms in Design

• Safety devices (e.g., airbags, seat belts) manage forces by prolonging the duration over which they act, reducing the peak force experienced by an occupant.

• By increasing time, the force is 'stretched,' leading to reduced impact and injury risk.

Application in Vehicle Collisions

• In vehicle accidents:

  • If a car hits a rigid wall (short stopping time), the resultant forces are high and dangerous.

  • With a deformable wall (like a soft wall), stopping time is increased, resulting in lower peak forces.

Protective Equipment

• Protective equipment is made from both hard and soft materials:

  • Hard shells spread the impact area (outer layer).

  • Softer, deformable layers extend stopping time, reducing peak force.

• Materials decompress to provide longer stopping time, thereby lowering the peak force on the body.

Impulse and Change in Momentum

• If a sudden change in momentum is needed without high forces, it is preferable to extend the time of force application.

Case Examples

• Gradual Braking: Gradual braking from a distance increases stopping time and reduces required friction force, enhancing control, especially on slippery surfaces.

• Baseball Batting: Players should swing through the ball to increase contact time, resulting in greater impulse and momentum change, leading to higher exit velocity.

Law of Conservation of Linear Momentum

Definition

• States that, in the absence of external forces, the total momentum of a system remains constant during collisions.

• Formula:

  • $\text{Initial momentum of Object 1} + \text{Initial momentum of Object 2} = \text{Final momentum of Object 1} + \text{Final momentum of Object 2}$.

• Individual momenta may change, but the total momentum before and after the collision stays the same.

Equations for Conservation of Momentum

• For object masses and velocities:

  • Object 1: mass = $m1$, initial velocity = $v1$, final velocity = $v_1'$.

  • Object 2: mass = $m2$, initial velocity = $v2$, final velocity = $v_2'$.

• General formula:

  • $m<em>1 v</em>1 + m<em>2 v</em>2 = m<em>1 v</em>1' + m<em>2 v</em>2'$.

Types of Collisions

1. Elastic Collisions

   • Objects remain separate and bounce apart after the collision (e.g., billiards, kicking a football).

   • Momentum is conserved as described above.

2. Inelastic Collisions

   • Objects stick together and move as one unit after the collision (e.g., car crashes, tackling, catching).

   • For perfectly inelastic collisions, total momentum equation is:

   • $m<em>1 v</em>1 + m<em>2 v</em>2 = (m<em>1 + m</em>2) v_{final}$.

Example Problem: Inelastic Collision of Two Cars

• Scenario:

  • Car 1: mass = 150 kg, velocity = 40 m/s (north to south).

  • Car 2: mass = 200 kg, velocity = 60 m/s (south to north).

  • Both cars collide and move together post-collision.

• Determine common final velocity:

  • Use inelastic collision formula:

  • $150 imes 40 + 200 imes 60 = (150 + 200) v_{final}$.

  • Solve for $v<em>{final}$, yielding $v</em>{final} = 51.42 m/s$; direction is north (because the southern car had more momentum).

• The direction of movement post-collision depends on relative momenta.

Important Points

• Direction post-collision is determined by relative momentum, not solely mass or individual velocity.

Conclusion

• The applications of impulse and conservation of momentum are critical to understanding dynamics in various real-world scenarios, from sports to automotive safety. Understanding how to manipulate these physical concepts aids in designing safer systems and improving performance across different activities.