Two Dimensional Collisions Worked Examples Cheat Sheet

Key Facts

• Momentum is a vector, so for each object $\vec{p} = m\vec{v}$ and its components are $p_x = mv_x$ and $p_y = mv_y$.
• In every isolated collision, total momentum is conserved in both directions: $\sum p_{x,i} = \sum p_{x,f}$ and $\sum p_{y,i} = \sum p_{y,f}$.
• Velocity components from an angle $\theta$ measured from the positive $x$ direction are $v_x = v\cos\theta$ and $v_y = v\sin\theta$.
• Speed and direction can be rebuilt from components using $v = \sqrt{v_x^2 + v_y^2}$ and $\theta = \tan^{-1}\left(\frac{v_y}{v_x}\right)$.
• For a perfectly inelastic collision where objects stick together, use $m_1\vec{v}_{1i} + m_2\vec{v}_{2i} = (m_1 + m_2)\vec{v}_f$.
• For an elastic collision, kinetic energy is also conserved, so $\frac{1}{2}m_1v_{1i}^2 + \frac{1}{2}m_2v_{2i}^2 = \frac{1}{2}m_1v_{1f}^2 + \frac{1}{2}m_2v_{2f}^2$.
• If one object is initially at rest, its initial momentum components are $p_{x,i} = 0$ and $p_{y,i} = 0$ for that object.
• A correct two dimensional solution must satisfy both component momentum equations, not just the equation in the direction of the initial motion.

Vocabulary

Momentum

Momentum is the vector quantity $\vec{p} = m\vec{v}$ that measures an object's mass in motion.

Impulse

Impulse is the change in momentum, written $\vec{J} = \Delta \vec{p}$.

Elastic collision

An elastic collision is a collision in which both momentum and kinetic energy are conserved.

Inelastic collision

An inelastic collision is a collision in which momentum is conserved but kinetic energy is not conserved.

Component

A component is the part of a vector along one axis, such as $v_x$ or $v_y$.

Isolated system

An isolated system has no net external force, so its total momentum remains constant.