Collisions and Momentum Conservation

The Law of Conservation of Momentum

When there are no external forces acting on a system, it is called an isolated system. When a collision occurs in an isolated system, momentum is conserved. The law of conservation of momentum tells us that momentum is constant before and after the collision. So, while mechanical energy is not always conserved during a collision, momentum in an isolated system is always conserved. As mentioned above, the forces exerted by the objects on each other are much greater than the external forces. So, for the examples we will consider, we will ignore any external forces and consider the system as isolated.

Formulas for Collisions with Momentum Conservation

Using the laws of conservation of momentum and energy, we can write down equations to describe the motion of objects in a collision. Here, we will focus on linear momentum, although this concept can also be applied in the form of angular momentum. Let’s consider separately the two types of collisions we mentioned above: elastic collisions and inelastic collisions.

Formula for Elastic Collisions

For elastic collisions, both mechanical energy and momentum are conserved. Thus we have an equation that we can write for each of these. The conservation of mechanical energy tells us that the energy before the collision,$E_i$, must equal the energy after the collision,$E_f$:

$E_i=E_f$

Billiard balls colliding is an example of an elastic collision, Pixabay

The mechanical energy is the sum of the kinetic and potential energy of the system, but for now let’s consider a system that has just kinetic energy, like a billiard ball about to collide with another billiard ball. We will assume that both billiard balls are moving in a straight line and the collision is one-dimensional. The total energy before the collision is given by

$$\begin{gathered} E_i=K_{1i}+K_{2i} \\ =\frac{1}{2}{m}_1{v_{1i}}^2+\frac{1}{2}{m}_2{v_{2i}}^2, \end{gathered}$$

where$m_1$and $m_2$are the masses of the billiard balls, and$v_{1i}$and$v_{2i}$are their initial velocities. Billiard balls typically have the same mass, but for generality we will assume they are slightly different. The total energy after the collision is given by

$$\begin{gathered} E_f=K_{1f}+K_{2f} \\ =\frac{1}{2}{m}_1{v_{1f}}^2+\frac{1}{2}{m}_2{v_{2f}}^2, \end{gathered}$$

where$v_{1f}$and$v_{2f}$are the final velocities of the billiard balls. Equating these gives us:

$\begin{array}{rcl}\frac{1}{2}{m}_1{v_{1i}}^2+\frac{1}{2}{m}_2{v_{2i}}^2& =& \frac{1}{2}{m}_1{v_{1f}}^2+\frac{1}{2}{m}_2{v_{2f}}^2\\ & & \\ \bcancel{\frac{1}{2}}{m}_1{v_{1i}}^2+\bcancel{\frac{1}{2}}{m}_2{v_{2i}}^2& =& \bcancel{\frac{1}{2}}{m}_1{v_{1f}}^2+\bcancel{\frac{1}{2}}{m}_2{v_{2f}}^2\\ & & \\ m_1{v_{1i}}^2+m_2{v_{2i}}^2& =& m_1{v_{1f}}^2+m_2{v_{2f}}^2.\end{array}$

This is the equation we have arrived at using the conservation of mechanical energy. Now, let’s look at the equation we can get from the conservation of momentum. The initial linear momentum,$P_i$, must equal the final linear momentum, $P_f$:

$P_i=P_f$

The linear momentum before the collision is

and the linear momentum after the collision is

$$\begin{gathered} P_f=P_{1f}+P_{2f} \\ =m_1{v}_{1f}+m_2{v}_{2f}. \end{gathered}$$

We get our equation from the conservation of linear momentum by equating these:

$m_1{v}_{1i}+m_2{v}_{2i}=m_1{v}_{1f}+m_2{v}_{2f}$

These equations look like they have a lot of unknown variables, but most problems give us the mass and initial velocities of the objects in the system. Recall that whenever we have two equations, we can solve for at most two unknown variables. Thus, we can then use both the conservation of energy and the conservation of momentum equations to solve for the final velocity of each of the two colliding objects.

Formula for Inelastic Collisions

For inelastic collisions, the mechanical energy is not conserved so we cannot equate the initial and final mechanical energies. We can, however, still use the conservation of momentum. Let’s consider a one-dimensional situation similar to the example above, except this time when the balls collide, some kinetic energy is lost to other forms of energy like thermal and sound energy. The equation to describe the conservation of momentum is the same as before:

$\begin{array}{rcl}{P}_i& =& P_f\\ & & \\ P_{1i}+P_{2i}& =& P_{1f}+P_{2f}\\ & & \\ m_1{v}_{1i}+m_2{v}_{2i}& =& m_1{v}_{1f}+m_2{v}_{2f}\end{array}$

Since we only have one equation, we can only solve for one variable. We can do this if the problem gives us the values of the other variables, or we can solve for the ratio of the final velocities when given the values of the masses and initial velocities.

A special case for inelastic collisions is when the objects stick together after the collision. This is called a perfectly inelastic collision. In this case, they share the same final velocity. Imagine that one of the balls from the example above is sticky and sticks to the other ball after the collision. Equating the initial momentum and the final momentum would then give us:

$\begin{array}{rcl}{P}_i& =& P_f\\ & & \\ P_{1i}+P_{2i}& =& P_{1,2f}\\ & & \\ m_1{v}_{1i}+m_2{v}_{2i}& =(& m_1+m_2)v_f\end{array}$

For a perfectly inelastic collision, there is just one final velocity to find, which we can do if we are given the other variables.

Collisions and Momentum Conservation Examples

Since collisions happen frequently in our daily lives, we need to be able to set up energy and momentum conservation equations based on each situation. Below are a couple of examples to help you practice.