One-Dimensional Collisions — AP Physics 1
1. Fundamentals of One-Dimensional Collisions ★★☆☆☆ ⏱ 3 min
One-dimensional (1D) collisions are short-duration, high-force interactions where all motion before, during, and after the collision occurs along a single straight line. They are often called 'head-on collisions' on AP exams.
Because collisions are very short in duration, external forces like gravity and friction are negligible during the interaction. This means the total momentum of the closed system of colliding objects is always conserved, regardless of collision type. This topic makes up a significant portion of Unit 5 (Momentum), which accounts for 14–18% of your total AP Physics 1 score.
2. Conservation of Momentum and Collision Classification ★★☆☆☆ ⏱ 4 min
The core principle for all 1D collisions is conservation of momentum for a closed, isolated system: total momentum before collision equals total momentum after collision, provided no net external force acts. For two objects, this gives the general equation:
m_1 v_{1i} + m_2 v_{2i} = m_1 v_{1f} + m_2 v_{2f}
Where $m_1, m_2$ are object masses, $v_{1i}, v_{2i}$ are initial velocities, and $v_{1f}, v_{2f}$ are final velocities. Signs are critical: velocity is negative if it points opposite your chosen positive direction. Collisions are classified by their change in total kinetic energy:
- **Elastic collisions**: Both momentum and kinetic energy are conserved; no energy is lost to heat, sound, or deformation.
- **Inelastic collisions**: Momentum is conserved, but some kinetic energy is lost to other forms. Most real-world collisions fall into this category.
- **Perfectly inelastic collisions**: A special case of inelastic collision where objects stick together after collision, sharing the same final velocity $v_f$. For this case, the momentum equation simplifies to:
m_1 v_{1i} + m_2 v_{2i} = (m_1 + m_2) v_f
Exam tip: Always explicitly define your positive direction at the start of any 1D collision problem, even if the question does not ask you to. This prevents costly sign errors that AP graders will mark down.
3. Elastic One-Dimensional Collisions ★★★★☆ ⏱ 4 min
For elastic 1D collisions, we have two independent conservation equations: one for momentum, one for kinetic energy. Instead of solving a system of one linear and one quadratic equation (which is slow and error-prone on the exam), we can combine the two laws to get a simple linear relation between velocities. This relation gives the relative velocity rule:
v_{1i} - v_{2i} = v_{2f} - v_{1f}
This means the relative speed of approach before the collision equals the relative speed of separation after the collision. This rule is only valid for elastic 1D collisions, and cuts solving time by more than half.
Exam tip: If you forget the relative velocity rule, you can always derive it from the two conservation laws during the exam. Memorizing it saves significant time on multi-part problems.
4. Kinetic Energy Change in Collisions ★★★☆☆ ⏱ 3 min
AP Physics 1 regularly asks you to calculate or interpret the change in total kinetic energy during a 1D collision. While momentum is always conserved for an isolated system, the change in kinetic energy $\Delta KE = KE_{\text{total, }f} - KE_{\text{total, }i}$ defines the collision type:
- $\Delta KE = 0$: Elastic collision, no kinetic energy lost
- $\Delta KE < 0$: Inelastic collision, kinetic energy is lost to other forms (kinetic energy cannot increase in a collision between two free objects)
- Maximum possible kinetic energy loss always occurs in perfectly inelastic collisions, as objects stick together and have the minimum possible final total kinetic energy consistent with momentum conservation
Lost kinetic energy is converted to heat, sound, work done to deform objects, or stored as internal potential energy. To classify a collision, you must always calculate $\Delta KE$ explicitly, even if objects bounce apart.
Exam tip: Do not assume a collision is elastic just because the objects bounce off each other; most real bounces still lose some kinetic energy.
5. Concept Check ★★★☆☆ ⏱ 2 min
Common Pitfalls
Why: Students confuse speed (a scalar) with velocity (a vector) and forget 1D motion still requires signs to represent direction.
Why: Students memorize the time-saving relation and forget it only applies when kinetic energy is conserved.
Why: Students mix the form of momentum and kinetic energy equations, since both are additive for the system.
Why: Students confuse the two conservation laws and incorrectly generalize momentum conservation to kinetic energy.
Why: Students forget that sticking together means the two objects move at the same velocity.