Elastic and Inelastic Collisions
The momentum is always conserved. The kinetic energy is the question.
Describe the kinetic energy of a system before and after a collision, and classify the collision by what happened to it.
Everything on this page serves that one sentence.
Topic 4.3 gave you one equation that holds for every collision on a level track. That equation is not always enough. Two carts that collide have two velocities afterward, and one equation cannot find two unknowns. This topic supplies the second piece of information, and it is a statement about energy rather than momentum.
1. Elastic means the kinetic energy survives
A collision is elastic when the total kinetic energy of the system after the collision equals the total kinetic energy before it.
\(K_{\text{before}} = K_{\text{after}}\)elastic, and only elastic
Read that carefully, because it is a statement about the total. The kinetic energies of the individual objects are free to change, and in almost every elastic collision they do. One cart can leave with much more than it arrived with, as long as the other leaves with correspondingly less.
Perfectly elastic collisions between everyday objects do not quite exist. Two hard steel balls, or two gliders on an air track with magnets facing each other, come close enough that the model is worth using. A problem that calls a collision elastic is handing you the second equation, and that is what the word is doing there.
One collision, run again and again with one thing changed. Drag the bottom slider from sticking at the left to perfectly elastic at the right. The two arrows at the top are the total momentum before and after. The three bars underneath are the kinetic energy.
Test thisLeave the masses and the velocities alone. Move the bottom slider to the far right, then to the middle, then to the far left, and read the two momentum arrows and the three energy bars at each position.
Report what happenedSay what happened to the momentum across the three settings, and what happened to the kinetic energy. Give the energy numbers.
Now without the simulationIn which of these collisions is the total momentum of the two carts conserved?
Topic questions
1 Which statement defines an elastic collision?
2 A 2.0 kg cart moving at 4.0 m/s collides elastically with a stationary 2.0 kg cart. Afterward,
2. Inelastic means some of it is gone
A collision is inelastic when the total kinetic energy of the system decreases. That covers almost everything that happens in a real room.
The energy is not destroyed. Nonconservative forces during the contact transform it into forms that are not kinetic energy of the objects: the permanent bending of a bumper, the sound you hear, and a very small rise in temperature of both objects. None of it comes back, which is exactly what makes those forces nonconservative.
Momentum does not care. Every collision on this page conserves momentum, elastic or not. The energy question and the momentum question are separate, and the most common error in this unit is letting the answer to one decide the other.
Topic questions
3 Two carts collide and the total kinetic energy of the system afterward is less than before. Which statement is correct?
4 In an inelastic collision between two cars, the kinetic energy that is no longer kinetic energy of the cars has mostly become
3. Perfectly inelastic, where the most is lost
A collision is perfectly inelastic when the objects stick together and move off with the same velocity. That single sentence is worth as much as the elastic condition, because it supplies the missing equation just as cleanly: there is now one unknown velocity instead of two.
\(m_1\vec{v}_1 + m_2\vec{v}_2 = (m_1 + m_2)\vec{v}_f\)one velocity afterward
Of all the collisions that start from the same two carts moving in the same two ways, the perfectly inelastic one loses the most kinetic energy. Nothing can lose more, because the objects would then have to pass through each other. Move the slider on the first simulation all the way to the left and that is what the bottom bar is showing you.
The perfectly inelastic case on its own. Drag the sliders to set the two masses and the speed of the moving cart. The bar underneath divides the original kinetic energy into the part that is still kinetic afterward and the part that is not.
Test thisSet both masses to 2.0 kg and the speed to 4.0 m/s and read the fraction kept. Now leave A alone and make B much heavier, up to 6.0 kg, and read it again.
Report what happenedGive the fraction of the kinetic energy kept in both cases. Does hitting a heavier stationary object keep more of the energy or less, and does the momentum change at all between the two cases?
Now without the simulationA 2.0 kg cart moving at 4.0 m/s strikes a stationary 2.0 kg cart and the two stick together. The kinetic energy of the system afterward is
Topic questions
5 A 3.0 kg cart moving at 2.0 m/s strikes a stationary 1.0 kg cart and the two stick together. The kinetic energy of the system afterward is
6 Two carts approach each other and collide. Which outcome removes the most kinetic energy from the system?
4. Which one is in front of you
Nothing on an exam will label a collision for you unless the label is a piece of information you are meant to use. So read what you are given and sort it yourself.
If the objects end up moving together, it is perfectly inelastic, and momentum alone finishes the problem. If the word elastic appears, you have two equations and you may need both. If neither is true, you have one equation and you will be given something else, usually one of the final velocities.
The order of work. Momentum first, always, because it holds regardless. Then ask what happened to the kinetic energy, and only then reach for a second equation.
Topic questions
7 A problem states that two gliders collide and move off together. How many equations do you need to find their common final velocity?
8 Two carts collide on a level track. A student calculates that the total kinetic energy afterward is larger than before. The most likely explanation is that
That is the whole of Unit 4. The problem set and the practice quiz come next, and then the unit test, which will ask you to pick a system before it asks you to calculate anything.