Kinetics of Particles: Impulse and Momentum

Why it matters

Understanding the kinetics of particles through impulse and momentum is crucial for predicting and analyzing the motion of objects when subjected to forces over time. This knowledge is essential in designing safe and efficient structures and systems, such as vehicles and machinery, where dynamic forces are at play.

Key ideas

• Impulse: Impulse is the time integral of force; for constant force it equals force multiplied by duration. It causes a change in momentum of a particle.
• Momentum: Momentum is the product of the mass and velocity of a particle. It is a vector quantity, having both magnitude and direction.
• Impulse-Momentum Principle: This principle states that the change in momentum of a particle is equal to the impulse applied to it.
• Conservation of Momentum: In the absence of external forces, the total momentum of a system remains constant.

Formulas

• Impulse: J = F·Δt
  • J: Impulse (N·s)
  • F: Force (N)
  • Δt: Time duration (s)
• Momentum: p = m·v
  • p: Momentum (kg·m/s)
  • m: Mass (kg)
  • v: Velocity (m/s)
• Impulse-Momentum Principle: J = Δp
  • Δp: Change in momentum (kg·m/s)
• Conservation of Momentum: m₁·v₁ + m₂·v₂ = m₁·v₁' + m₂·v₂'
  • m₁, m₂: Masses of particles (kg)
  • v₁, v₂: Initial velocities (m/s)
  • v₁', v₂': Final velocities (m/s)

Worked example

Given: A 5 kg object moving at 10 m/s is subjected to a constant net force of 20 N in its initial direction of motion for 3 seconds.

1. Calculate the initial momentum.
   • p_initial = m·v = 5 kg · 10 m/s = 50 kg·m/s
2. Calculate the impulse.
   • J = F·Δt = 20 N · 3 s = 60 N·s
3. Apply the impulse-momentum principle.
   • J = Δp = p_final - p_initial
   • 60 N·s = p_final - 50 kg·m/s
   • p_final = 110 kg·m/s
4. Calculate the final velocity.
   • p_final = m·v_final
   • 110 kg·m/s = 5 kg · v_final
   • v_final = 22 m/s

Final Answer: 22 m/s

Common mistakes

• Confusing impulse with force; impulse is force applied over time.
• Forgetting that momentum is a vector and must consider direction.
• Ignoring external forces when applying conservation of momentum.

For GATE CE

Questions often involve calculating changes in velocity or force using the impulse-momentum principle. Practice problems involving collisions and systems with multiple particles to master conservation of momentum.

Quick check

1. What is the unit of impulse?
2. How does impulse relate to momentum?
3. What happens to momentum in an isolated system?

Answers: 1. N·s 2. Impulse equals change in momentum 3. It remains constant.