Kinetics of Particles: Impulse and Momentum

Why it matters

Impulse and momentum are fundamental concepts in engineering mechanics that help in analyzing the motion of particles when subjected to forces over time. These principles are crucial in designing safety features in vehicles, understanding collisions, and predicting the behavior of mechanical systems.

Key ideas

• Impulse: Impulse is the time integral of a force. For a constant force, it is the product of force and duration. It changes the momentum of a particle.
• Momentum: Momentum is the product of mass and velocity of a particle. It is a vector quantity and is conserved in isolated systems.
• 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.

Assumptions and general form

For a constant-mass particle observed in an inertial frame, J_net = ∫(ΣF_ext)dt = m(v_f − v_i). All quantities except time and mass are vectors. Integrate each component when forces vary with time; impulse is the signed area under the force–time graph. The formula FΔt uses either a constant force or the time-average force.

For a system of particles, total momentum changes by the net external impulse. Internal action–reaction impulses cancel. Momentum can be conserved in one direction even when an external impulse acts in another direction. Conservation of momentum does not generally imply conservation of kinetic energy during a collision.

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 = m·v_f - m·v_i
  • v_f: Final velocity (m/s)
  • v_i: Initial velocity (m/s)

Worked example

Given: A 5 kg object is initially moving at 2 m/s. A constant net force of 10 N acts in the direction of the initial velocity for 3 seconds. Take that direction as positive and assume the mass remains constant.

1. Calculate the initial momentum.
   • p_i = m·v_i = 5 kg · 2 m/s = 10 kg·m/s
2. Calculate the impulse.
   • J = F·Δt = 10 N · 3 s = 30 N·s
3. Apply the impulse-momentum principle to find the final velocity.
   • J = m·v_f - m·v_i
   • 30 N·s = 5 kg·v_f - 10 kg·m/s
   • 5 kg·v_f = 40 kg·m/s
   • v_f = 8 m/s

Final Answer: The final velocity is 8 m/s.

Common mistakes

• Confusing impulse with force; impulse is force applied over time.
• Forgetting that momentum is a vector and has direction.
• Ignoring the conservation of momentum in isolated systems.

For GATE ME

Questions often involve calculating the final velocity of a particle after an impulse or determining the impulse required to change a particle's velocity. Practice problems involving collisions and conservation of momentum.

Quick check

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

Answers: 1. N·s 2. Impulse changes momentum. 3. It remains constant.