Collisions in one dimension are events where two objects interact while moving along a single straight line. They are important because the total momentum of an isolated system stays constant, even when the objects bounce, stick, or exchange speeds. This makes collisions a powerful way to predict motion after an impact using information from before the impact.
Carts on a frictionless track are a classic model because outside forces along the track can be made very small.
The main idea is to choose a positive direction, assign signs to velocities, and write momentum conservation for the whole system. In an elastic collision, both momentum and kinetic energy are conserved, which gives enough information to solve for both final velocities. In a perfectly inelastic collision, the objects stick together and move with one common final velocity.
Real collisions often fall between these extremes, so understanding the ideal cases helps students analyze experiments and estimate outcomes.
Understanding Physics: Collisions in One Dimension
During an impact, each object pushes on the other for the same short time. The forces have equal size and opposite directions. This follows from Newton's third law.
The force may be very large, but its duration is usually tiny. Force times collision time is called impulse. Each object receives an impulse that changes its own momentum.
For the pair taken together, these internal changes cancel. This is why it is important to draw a boundary around the system before doing any calculation.
A track may have some friction, yet its impulse during a brief collision can be small enough to ignore. A long push from a hand or a wall is different because it gives the system a significant external impulse.
The word elastic describes what happens to kinetic energy during the impact. Even hard objects deform slightly when they collide. In a nearly elastic event, that deformation acts like a spring.
Energy is stored briefly, then returned to motion as the objects separate. In a perfectly inelastic event, the deformation does not fully reverse. Some kinetic energy becomes internal energy, sound, or permanent shape change.
This does not mean energy disappears. It means that less energy remains available as motion of the objects.
When objects stick, their shared final speed is controlled by how much momentum each object brought in. A heavy slow object can strongly affect the motion of a light fast object.
A one dimensional problem becomes much easier when every velocity has a sign from the start. Speed is always positive, while velocity can be positive or negative. An object that rebounds has changed the sign of its velocity.
Students often lose marks by treating a leftward velocity as a positive number in one line, then as a negative number in another. Keep one chosen direction for the whole problem. In an elastic collision, one equation comes from total momentum and a second relation comes from the kinetic energy condition or from the relative speed rule.
These two pieces of information determine the two unknown final velocities. For equal masses, a moving object striking an identical object at rest transfers its velocity to the second object in the ideal elastic case. Different masses give less familiar results, including rebounds of the lighter object.
You meet these ideas in pool, bowling, shopping carts, railway couplings, and crash testing. A pool ball collision is often close to elastic, although rolling friction and spin make a real table more complicated than a basic model. Car crashes are strongly inelastic because crumple zones are designed to deform.
The deformation increases collision time, which can reduce the average force on people inside. In classroom experiments, use motion sensors or video to compare values just before and after contact. Check units carefully.
Momentum has units of kilogram metres per second, while kinetic energy has units of joules. Finally, state which object or objects belong to the system and explain why outside impulses can be neglected. That assumption is the foundation of every useful collision calculation.
Key Facts
- Momentum of one object: p = mv
- Momentum conservation in 1D: m1v1i + m2v2i = m1v1f + m2v2f
- Kinetic energy: K = 1/2 mv^2
- Elastic collision condition: 1/2 m1v1i^2 + 1/2 m2v2i^2 = 1/2 m1v1f^2 + 1/2 m2v2f^2
- Perfectly inelastic final velocity: vf = (m1v1i + m2v2i) / (m1 + m2)
- 1D elastic relative speed rule: v1i - v2i = -(v1f - v2f)
Vocabulary
- Momentum
- Momentum is the product of an object's mass and velocity, and it includes direction.
- Impulse
- Impulse is the change in momentum caused by a force acting over a time interval.
- Elastic collision
- An elastic collision is a collision in which total momentum and total kinetic energy are both conserved.
- Inelastic collision
- An inelastic collision is a collision in which total momentum is conserved but kinetic energy is not conserved.
- Perfectly inelastic collision
- A perfectly inelastic collision is one where the objects stick together and move with the same final velocity.
Common Mistakes to Avoid
- Ignoring velocity signs, which is wrong because momentum is a vector and direction matters in one dimension.
- Assuming kinetic energy is always conserved, which is wrong because only elastic collisions conserve total kinetic energy.
- Using mass in grams instead of kilograms, which gives incorrect SI units and can make momentum and energy calculations inconsistent.
- Solving each object separately without using the system, which is wrong because momentum conservation applies to the combined isolated system during the collision.
Practice Questions
- 1 A 2.0 kg cart moving at +3.0 m/s collides with a 1.0 kg cart initially at rest. If they stick together, what is their final velocity?
- 2 A 0.50 kg cart moving at +4.0 m/s elastically collides with a 0.50 kg cart initially at rest. What are the final velocities of both carts?
- 3 Two carts collide on a nearly frictionless track. After the collision, the total kinetic energy is smaller than before, but the total momentum is unchanged. What type of collision is this, and why can momentum still be conserved?