Kinetics of Particles: Impulse and Momentum

Kinetics of Particles: Impulse and Momentum

The method of impulse and momentum is used to solve problems involving force, time, and velocity. It is particularly useful for impulsive forces (collisions).

Principle of Linear Impulse and Momentum

The integral of Newton's Second Law with respect to time yields:

mv1+∫t1t2F dt=mv2m \mathbf{v}_1 + \int_{t_1}^{t_2} \mathbf{F} \, dt = m \mathbf{v}_2

Or simply: L1+I1−2=L2\mathbf{L}_1 + \mathbf{I}_{1-2} = \mathbf{L}_2

Where:

  • Linear Momentum: L=mv\mathbf{L} = m \mathbf{v}
  • Linear Impulse: I=∫F dt\mathbf{I} = \int \mathbf{F} \, dt

Conservation of Linear Momentum

If the sum of external impulses acting on a system of particles is zero, the total linear momentum remains constant. ∑mv1=∑mv2\sum m \mathbf{v}_1 = \sum m \mathbf{v}_2 This is common in collision problems where external forces (like gravity) are negligible during the short impact time.

Impact

Impact occurs when two bodies collide during a very short time interval.

  • Coefficient of Restitution (ee): Measures the elasticity of the impact. e=(vB)2−(vA)2(vA)1−(vB)1e = \frac{(v_B)_2 - (v_A)_2}{(v_A)_1 - (v_B)_1}
    • e=1e = 1: Perfectly elastic (energy conserved).
    • e=0e = 0: Perfectly plastic (bodies stick together).

Example: Collision

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