The types of one dimensional collisions we solve fall into several categories.
Not every possible collision falls into one of the following three categories, but this might help with many collision problems.
All of the collisions use the same concept: Law of Conservation of Momentum. Total momentum before collisions equals total momentum after the collision.
“Bounce” Collisions
(perfectly elastic or partially elastic collisions)
The 1 and 2 subscripts are for object 1 and object 2.
The “prime” (apostrophe) marks for the velocities on the right side are for the velocity at a later time, after the collision. Both objects will have a different velocity before and after the collision.
This equation is the most general equation. It can work for any collision between two objects, but there are shortcuts that you can take in some cases. For example…
“Hit and Stick” Collisions (Perfectly Inelastic)
When the two objects stick together instead of bouncing, this makes the math easier, because both objects have the same velocity at the end.
That means that instead of v1‘ and v2‘, both are just v’.
After collecting the common v terms, the equation looks like this:
On the right side (after the collision), the two objects combine and act like a single object with one velocity, and with the total mass of both objects combined.
“Hit and STOP” Collisions
In this case, the math is simpler than a hit and stick equation since the final velocity of both objects is zero. This means the entire right side of the equation equals zero:
“Explosion” Collisions with both objects starting at rest.
This type of collision happens when a cannon fires. The cannon ball goes forward, and the cannon goes backwards (recoils). (This also happens when objects are simply pushing apart from each other – no actual explosion required!)
If both objects start out at rest, the left side of the equation is zero. Both momentum terms are zero since the masses are multiplied by zero velocity.
Object 1 has the same amount of momentum as Object 2 at the end, but in the opposite direction. So for example, if each object had 500 kgm/s of momentum, it would add up like this:
0 = +500 + (− 500)
Also note that the biggest object will have a smaller change in velocity. The smaller object needs to be going faster for them to have equal momentum.
Explosion Collision with the objects initially moving and stuck together
In this case, you can combine the masses together on the left side. Both objects have the same velocity, v, since they are stuck together.
