Topic 4.4 Collisions: analyze elastic and inelastic collisions using conservation of momentum, and distinguish them by whether kinetic energy is conserved.

What this topic is asking

The College Board (Topic 4.4) wants you to analyze collisions using conservation of momentum, and to classify them as elastic or inelastic by asking whether kinetic energy is conserved. Momentum is conserved in every collision with no external force; kinetic energy is conserved only in elastic collisions. A perfectly inelastic collision, in which the objects stick together, loses the most kinetic energy.

Momentum is always conserved; kinetic energy is not

This split is the heart of the topic. Momentum conservation always gives you one equation (per dimension). Whether you get a second equation depends on the collision type: elastic collisions add kinetic-energy conservation, while inelastic ones do not, so you need extra information (such as the objects sticking together).

The three types of collision

Most real collisions are inelastic to some degree. A car crash, a ball of clay hitting a wall, and two railway carriages coupling are inelastic; the coupling case is perfectly inelastic. A perfectly inelastic collision is the easiest to solve, because the single common final velocity is found from momentum conservation alone.

Solving collision problems

The reliable routine is to start with momentum conservation and then decide what else you know:

• For a perfectly inelastic collision, the objects share one final velocity, so momentum conservation alone gives it: $m_1 v_1 + m_2 v_2 = (m_1 + m_2)v_f$.
• For an elastic collision, write both momentum conservation and kinetic-energy conservation, giving two equations for the two unknown final velocities.
• For a general inelastic collision, you usually need additional data (such as one final velocity) since kinetic energy is not conserved.

To check the type after solving, compare the total kinetic energy before and after: if it is unchanged the collision is elastic, if it drops it is inelastic, and the amount lost is the mechanical energy converted to heat and sound. This connects directly to the energy bookkeeping of Topic 3.4: the kinetic energy "lost" in an inelastic collision is not destroyed, it becomes thermal and other internal energy, so total energy is still conserved even though mechanical energy is not. Two-dimensional collisions (such as glancing impacts) are handled by conserving momentum in each direction separately, since momentum is a vector. The strategic insight for the exam is that momentum conservation is the universal first step in every collision, and the question's wording, "stick together", "bounce elastically", "lose energy", tells you whether and how to bring in kinetic energy.

Try this

Q1. A $3.0$ kg cart at $2.0$ m/s collides with and sticks to a $1.0$ kg cart at rest. Calculate the common final speed. [2 points]

• Cue. $p_i = (3.0)(2.0) = 6.0$; $p_f = (4.0)v$, so $v = 6.0/4.0 = 1.5$ m/s.

Q2. In a collision the total kinetic energy before is $20$ J and after is $20$ J. State whether the collision is elastic or inelastic. [1 point]

• Cue. Kinetic energy is unchanged, so the collision is elastic.