Vehicle Collisions

Momentum is defined as the product of an object's speed and velocity (denoted $mv$). In an elastic collision between two (or more) objects, the total momentum of the system remains the same before and after the collision. This concept can be made more intuitive with an equation, which is written as follows:

$\sum{m_iv_i} = \sum{m_fv_f}$

However, if there are only two vehicles involved in the collision, this can be rewritten as

$m_{1}v_{i1} + m_{2}v_{i2} = m_{1}v_{f1} + m_{2}v_{f2}$

Usually either one of the velocities or the mass of one of the objects will be unknown. Henceforth problems involving collisions tend to involve some algebra.

Here is an example.

A $4500$-kg racecar initially moving south at $85$ miles per hour down a straight highway collides into a 16-wheeler truck initially at rest. After the collision, the car moves south at $24$ miles per hour while the truck is pushed south at $8$ miles per hour. What is the mass of the truck?

Solution: This is a two-vehicle collision, so the equation

$m_{1}v_{i1} + m_{2}v_{i2} = m_{1}v_{f1} + m_{2}v_{f2}$

will apply. Without loss of generality, let vehicle 1 be the car and let vehicle 2 be the truck. We are asked for the mass of the truck, which is denoted as $m_2$. Thus we want to solve the momentum equation for $m_2$.

Isolate all terms involving $m_2$. This will require two steps. First, subtract $m_2v_{f2}$ from both sides of the equation:

$m_{1}v_{i1} + m_{2}v_{i2} - m_2v_{f2} = m_{1}v_{f1}$

Now subtract $m_{1}v_{i1}$ from both sides of the equation:

$m_2v_{i2} - m_2v_{f2} = m_1v_{f1} - m_{1}v_{i1}$

Now that all terms with $m_2$ are on one side of the equation, factor out $m_2$ from the left-hand side.

$m_2(v_{i2} - v_{f2}) = m_1v_{f1} - m_{1}v_{i1}$

Isolate $m_2$ as follows:

$m_2 = \frac{m_1v_{f1} - m_1v_{i1}}{v_{i2} - v_{f2}}$

This form of the equation allows us to plug and chug to find $m_2$. All other variables have been given to us.

$m_2 = \frac{4500 * (-24) - 4500 * (85)}{0 - (8)} \Rightarrow$ $m_2 = \frac{-490500}{-8} \Rightarrow$ $m_2 = 61312.5 \; kg$

This is the mass of the truck, as desired.

The key to understanding vehicle collisions is the law of conservation of momentum. Momentum is conserved in all collisions; in other words, the momentum of the system before the collision is the same as the momentum after the system.

As a formula, this is expressed as:

$m_{1}v_{1i} + m_{2}v_{2i} = m_{1}v_{1f} + m_{2}v_{2f}$

Here we can see that the carts are travelling with an initial speed before they collide and bounce off of each other with an altered final speed. The carts could also collide while traveling in the same direction and the same formula would still apply. Remember to account for the direction in which the carts are travelling! Force is a vector and therefore has direction.

The energy of the system can also be calculated using the law of the conservation of energy:

${1}/{2}m_1v_{1i}^2 + {1}/{2}m_2v_{2i}^2 = {1}/{2}m_1v_{1f}^2 + {1}/{2}m_2v_{2f}^2$ + thermal energy lost in collision

If the thermal energy lost in the collision is 0, the collision is called elastic. If there is thermal energy lost in the collision, the collision is called inelastic.

Here is an example of an inelastic collision:

If the two moving objects stick together, this usually indicates that the collision was inelastic. Kinetic energy is not conserved.

Here is an example of an elastic collision:

Here, the kinetic energy is conserved. The truck is initially moving at a certain speed and moves slower after the collision and the car has an initial speed of zero and moves significantly faster after the collision. The truck's larger mass also factors into the equation and therefore the collision is elastic.

Collisions can also look like this:

Here, you must add the vector components of the car's momentums.

Momentum is a vector quantity that is often described as *mass in motion. * Momentum depends on both mass and velocity and is given by the equation: $\overrightarrow {p} = m * \overrightarrow{v}$
where momentum (p) is measured in kg m/s in the same direction as the velocity vector, mass (m) is measured in kg and velocity is measured in m/s with a direction.
There are two common types of collisions: $\\$ • Elastic collisions - Two or more objects collide and both kinetic energy and momentum are conserved in an isolated system ($F_{net} = 0$) . $\\$ • Inelastic collisions - Two or more objects collide and while momentum is conserved, kinetic energy is not conserved ($F_{net} = 0$).

During both types of collisions momentum is conserved when there are no external forces added to the system. This means that the momentum before a collision is equal to the momentum after a collision in the absence of the