Collisions in physics come in three flavors:

1. elastic collisions, where the kinetic energy of a system is conserved
2. inelastic collisions, where kinetic energy is not conserved
3. cherry

Okay, so two flavors. But the distinction between those two is important—what type of crash occurs in our problem determines what equations we can use to solve it.

The amount of kinetic energy lost as heat, sound, etc. in an inelastic collision is related to the coefficient of restitution (C_R), a ratio of the speeds of two objects before and after impact. C_R is always a number between 0 (maximum loss of kinetic energy, occurring when objects stick together after the collision) and 1 (no loss of kinetic energy—an elastic collision), and is given by:



Where the subscript "1'' indicates the first object, "2'' indicates the second, "i'' is the initial velocity before the collision, and "f'' is the final velocity after the collision.

In the case of both elastic and inelastic collisions, momentum is always conserved and becomes a useful tool for solving any problem dealing with collisions. For elastic collisions, our two conservation laws—momentum and energy—give us enough equations to solve for the velocities of both colliding objects.

Here's an example: two blocks, one weighing m₁ and the other weighing m₂, are sliding towards each other at v_1,i and v_2,i, respectively.



After the blocks collide, their masses don't change, but their velocities do, so that block 1 is now traveling at v_1,f and block 2 at v_2,f. Applying the laws of conservation of momentum and energy gives us this system of equations:

m₁v_1,i + m₂v_2,i = m₁v_1,f + m₂v_2,f



And solving these equations for v_1,f and v_2,f gives:



And if you thought that was a lot of algebra, well, you'd be right. Sorry about that.

Here’s another example: If Indiana Jones isn't quick enough the boulder will collide with him in an inelastic collision. Indiana Jones weighs m_ij and is running unsuccessfully away at v_ij,i. The boulder weighs m_b and is rolling at a constant v_b,i. If they collide, Indy sticks to the boulder, and thus their final velocities are the same at v_f. Conservation of momentum gives us the following:

m_ijv_ij,i + m_bv_b,i = m_ijv_f + m_bv_f

Common Mistakes

If an object is stationary, all of these formulas still apply—a stationary object just has a velocity of (you guessed it) zero.

When you calculate total momentum, stationary objects contribute zero, but can gain momentum if they start moving. Velocity is directional, so sometimes v₁ or v₂ will be negative, and in order to calculate total momentum you must subtract. Two identical cars driving away from each other at the same speed, for example, have a net momentum of zero.

Brain Snack

In an inelastic collision, kinetic energy may not be conserved, but total energy definitely still is. A lot of it is just transformed into sound or heat—or the total mechanical destruction of an otherwise perfectly solid object.