🚀 Impulse–Momentum Relations

Understanding Motion Through Impulse and Momentum 

The impulse–momentum principle provides a direct relationship between the impulse of a force and the resulting change in momentum of a particle or system.

For a particle, the impulse–momentum relation connects the initial and final states without requiring the complete force or acceleration history to be solved explicitly.

The animations in this section use representative problems to demonstrate how linear and angular impulse–momentum relations are formulated and applied.

Each problem connects the physical motion, applied forces and moments, momentum, and the resulting change in motion.

🧮 Governing Principles

Linear Impulse–Momentum

For a particle moving from state a to state b,

ΔG = ∫ F dt

where G = m v is the linear momentum.

The impulse of the resultant force therefore produces the corresponding change in linear momentum.

If the resultant external impulse is zero in a particular direction, the corresponding component of linear momentum is conserved:

ΔG = 0 → G = constant

This provides the basis for the conservation of linear momentum principle. For a system of particles, internal impulsive forces cancel in pairs, so the total linear momentum is conserved when the net external impulse is zero.

Angular Impulse–Momentum

Taking moments about a fixed point O,

ΔHₒ = ∫ Mₒ dt

where Hₒ is the angular momentum about O.

The angular impulse of the resultant moment therefore produces the corresponding change in angular momentum.

If the resultant external moment about a fixed point is zero over a time interval, the angular momentum about that point is conserved:

ΔHₒ = 0 → Hₒ = constant

This provides the basis for the conservation of angular momentum principle.

The correct application of the impulse–momentum principle requires identification of:

• the system or particle being analysed,

• the initial and final states,

• the relevant linear or angular momentum,

• the forces or moments producing the impulse, and

• any kinematic constraints relating the motion of different components.

🎥 Problems Explored 

1. 🚀 Particle Deflected by a Time-Varying Force

Problem: A particle moves initially with constant velocity on a frictionless horizontal plane. A force acts on it over a finite time interval, changing its momentum and consequently its direction of motion.

Concept: Application of the linear impulse–momentum relation to a particle subjected to a time-varying force.

🔍 Watch For:

• How the particle initially moves with constant momentum.

• How the applied force produces an impulse over the interval ta ≤ t ≤ tb.

• How the area under the force–time curve represents the accumulated impulse.

• How the particle's momentum changes progressively as the impulse accumulates.

• How the final momentum determines the new direction of motion.

• Observe the live comparison between the accumulated impulse, change in momentum, and the current momentum components.

2. 🌀 Particle Attached to a Spring — Angular Momentum

Problem: A particle moves on a frictionless horizontal surface while connected to a spring fixed at O. At position A, the particle has a velocity of 4 m/s. Determine its velocity as it passes position B.

Concept: Application of the angular impulse–momentum relation about a fixed point when the resultant moment of the force about that point is zero.

🔍 Watch For:

• How the spring force always acts along the line joining O and the particle.

• Why the moment of the spring force about O is zero.

• How the angular impulse about O therefore remains zero.

• How the angular momentum H₀ = mrv sin φ remains constant as the particle moves from A to B.

• How the changing spring length and velocity direction affect the angular momentum terms.

• Observe the live verification showing that the change in angular momentum is equal to the angular impulse about O.

The animation also shows the instantaneous spring length and the angle φ between the radial direction and the velocity vector.

Source: Meriam, Kraige & Bolton, Engineering Mechanics: Dynamics, 8th edition, Problem 3/228.