Momentum & Impulse

Linear Momentum

$\vec{p} = m\vec{v}$ Momentum is a vector quantity with units of $\text{kg·m/s}$.

Impulse

Impulse is the change in momentum: $\vec{J} = \vec{F}_{\text{avg}} \cdot \Delta t = \Delta \vec{p}$

This is the impulse-momentum theorem.

Conservation of Momentum

In an isolated system (no external net force): $\vec{p}_{\text{total, before}} = \vec{p}_{\text{total, after}}$ $m_1 v_{1i} + m_2 v_{2i} = m_1 v_{1f} + m_2 v_{2f}$

Types of Collisions

• Perfectly inelastic: objects stick together after collision. $m_1 v_{1i} + m_2 v_{2i} = (m_1 + m_2)v_f$

• Elastic (1D): Use both momentum conservation and $v_{1i} - v_{2i} = -(v_{1f} - v_{2f})$.

Center of Mass

$x_{\text{cm}} = \frac{m_1 x_1 + m_2 x_2}{m_1 + m_2}$

The center of mass of an isolated system moves with constant velocity.

Problem: A $3\ \text{kg}$ cart moving at $4\ \text{m/s}$ collides with and sticks to a $1\ \text{kg}$ cart at rest. Find the final velocity and the kinetic energy lost.

Solution:

Conservation of momentum: $3(4) + 1(0) = (3+1)v_f$ $v_f = \frac{12}{4} = 3\ \text{m/s}$

KE before: $\frac{1}{2}(3)(4)^2 = 24\ \text{J}$

KE after: $\frac{1}{2}(4)(3)^2 = 18\ \text{J}$

KE lost: $24 - 18 = 6\ \text{J}$ (converted to thermal energy, sound, deformation)

• Momentum is conserved in all collisions when no external net force acts.
• Impulse equals the change in momentum: $J = F\Delta t = \Delta p$.
• Kinetic energy is conserved only in perfectly elastic collisions.
• In perfectly inelastic collisions, objects stick together and maximum KE is lost.