Impulse and Momentum

Impulse and Momentum

Momentum is a measure of the motion of an object, taking into account both its mass and velocity. It is a fundamental concept for understanding collisions and explosions.

Linear Momentum and Impulse

Linear Momentum ($\mathbf{p}$)

Linear momentum is defined as the product of mass and velocity. $\mathbf{p} = m \mathbf{v}$ It is a vector quantity pointing in the direction of velocity. The unit is kg$\cdot$m/s.

Impulse ($\mathbf{J}$)

Impulse is the change in momentum resulting from a force acting over a time interval. $\mathbf{J} = \int_{t_i}^{t_f} \mathbf{F} \, dt = \mathbf{F}_{avg} \Delta t$

Impulse-Momentum Theorem: The impulse applied to an object equals its change in momentum. $\mathbf{J} = \Delta \mathbf{p} = \mathbf{p}_f - \mathbf{p}_i$

Conservation of Momentum

Principle of Conservation of Linear Momentum: If the net external force acting on a system is zero, the total linear momentum of the system remains constant. $\sum \mathbf{p}_i = \sum \mathbf{p}_f$ $m_1 \mathbf{v}_{1i} + m_2 \mathbf{v}_{2i} = m_1 \mathbf{v}_{1f} + m_2 \mathbf{v}_{2f}$

This principle applies to collisions and explosions.

Collisions

Collisions are interactions where objects exert relatively large forces on each other for a short time.

Elastic Collisions

Both momentum and kinetic energy are conserved. $\sum \mathbf{p}_i = \sum \mathbf{p}_f$ $\sum KE_i = \sum KE_f$

Inelastic Collisions

Momentum is conserved, but kinetic energy is not (some energy is lost to heat, sound, or deformation).

• Perfectly Inelastic Collision: The objects stick together after collision and move with a common velocity.