8.3 Collisions in One and Two Dimensions

Types of Collisions

Elastic vs. Inelastic Collisions

Every collision conserves momentum. What separates collision types is what happens to kinetic energy.

Elastic collisions conserve both momentum and total kinetic energy. The objects bounce off each other with no permanent deformation. Billiard ball impacts and collisions between gas molecules are close approximations. In practice, perfectly elastic collisions only truly occur at the atomic and subatomic scale.

Inelastic collisions conserve momentum but not kinetic energy. Some kinetic energy gets converted into heat, sound, or permanent deformation of the objects. Most real-world collisions (car crashes, a baseball hitting a bat) are inelastic to some degree.

Perfectly inelastic collisions are the extreme case: the objects stick together after impact and move as a single combined mass. This produces the maximum possible kinetic energy loss while still conserving momentum. A bullet embedding in a block of wood is a classic example.

One-Dimensional Collision Problem-Solving

In 1D, all motion happens along a single line, so you don't need vectors. Here's how to approach these problems by collision type.

For all 1D collisions, start with conservation of momentum:

$m_1 v_1 + m_2 v_2 = m_1 v_1' + m_2 v_2'$

where primed quantities ($v_1'$, $v_2'$) are post-collision velocities.

Elastic collisions give you a second equation because kinetic energy is also conserved:

$\frac{1}{2}m_1 v_1^2 + \frac{1}{2}m_2 v_2^2 = \frac{1}{2}m_1 v_1'^2 + \frac{1}{2}m_2 v_2'^2$

Solving these two equations simultaneously can get messy. A much faster approach uses the relative velocity reversal property, which is mathematically equivalent to the kinetic energy equation but far easier to work with:

$v_1 - v_2 = -(v_1' - v_2')$

This says the relative speed of approach equals the relative speed of separation. Use this alongside momentum conservation to solve for both final velocities without dealing with squared terms.

Perfectly inelastic collisions are actually the simplest to solve. The two objects share a single final velocity $v_f$:

$m_1 v_1 + m_2 v_2 = (m_1 + m_2) v_f$

You can then find the kinetic energy lost:

$\Delta KE = \frac{1}{2}m_1 v_1^2 + \frac{1}{2}m_2 v_2^2 - \frac{1}{2}(m_1 + m_2)v_f^2$

Coefficient of restitution ($e$) provides a single number that characterizes how "bouncy" a collision is:

$e = \frac{v_2' - v_1'}{v_1 - v_2}$

• $e = 1$: perfectly elastic (no kinetic energy lost)
• $e = 0$: perfectly inelastic (objects stick together)
• $0 < e < 1$: partially inelastic (most real collisions)

Step-by-step approach for any 1D collision:

1. Draw a diagram. Define a positive direction and assign signs to all velocities.
2. Write the momentum conservation equation.
3. Determine the collision type. If elastic, add the relative velocity reversal equation. If perfectly inelastic, set $v_1' = v_2' = v_f$.
4. Solve the system of equations for the unknowns.
5. Check your answer: Does momentum before equal momentum after? If elastic, does kinetic energy also balance?

Two-Dimensional Collisions

Vector Analysis of Two-Dimensional Collisions

When objects collide at angles (like a cue ball hitting another ball off-center), you need to work in two dimensions. The core idea is the same: momentum is conserved. But now momentum is a vector, so you conserve it independently in each component direction.

$x\text{-direction: } m_1 v_{1x} + m_2 v_{2x} = m_1 v_{1x}' + m_2 v_{2x}'$

$y\text{-direction: } m_1 v_{1y} + m_2 v_{2y} = m_1 v_{1y}' + m_2 v_{2y}'$

Step-by-step approach for 2D collisions:

1. Set up a coordinate system. Typically, align the x-axis with the initial velocity of one object. This often makes several initial components zero, which simplifies things considerably.
2. Break all velocity vectors into x and y components using trigonometry ($v_x = v\cos\theta$, $v_y = v\sin\theta$).
3. Write momentum conservation for x and y separately.
4. If the collision is elastic, add the kinetic energy conservation equation (this uses speeds, not components, so no extra vector work).
5. Solve the system of equations for the unknowns.
6. If you need final speeds and angles, recombine components: $v = \sqrt{v_x^2 + v_y^2}$ and $\theta = \tan^{-1}(v_y / v_x)$.

A common setup is a glancing collision, where one object is initially at rest and the incoming object deflects at some angle. After the collision, both objects move at different angles relative to the original direction. You'll typically be given enough information (masses, initial speed, one deflection angle) to solve for the rest.

Conservation Principles in Collision Problems

The vector form of momentum conservation applies to every collision, regardless of type or dimension:

$\vec{p}_1 + \vec{p}_2 = \vec{p}_1' + \vec{p}_2'$

For elastic 2D collisions, you get three equations total: x-momentum, y-momentum, and kinetic energy. That means you can solve for up to three unknowns (for example, two final speeds and one angle).

Center of mass frame. In the center of mass (CM) reference frame, the total momentum of the system is zero by definition. This simplifies collision analysis because the two objects always move in exactly opposite directions. In elastic collisions viewed from the CM frame, each object simply reverses its velocity. This frame is especially useful in particle physics, where collisions are analyzed in the CM frame and then transformed back to the lab frame.