Describe the collision between two vehicles of equal mass moving toward each other at equal speeds.

Answer 1

Hi Kasyah!

I'll do a quick run through of this scenario!

So let's say that "to the right" is the "positive" direction, and "to the left" is the "negative" direction.

Both cars have the same mass, which I will call m_car, and they both have the same speed, which I'll call v_car.

The initial momentum of the first car going to the right would then be:

p_1i = m_car*v_car

The initial momentum of the second car going to the left would then be:

p_2i = - m_car*v_car

So, let's set up a conservation of momentum equation...

p_1i + p_2i = p_1f + p_2f

( m_car*v_car ) + ( - m_car*v_car ) = p1f + p2f

0 = p_1f + p_2f

So the total momentum of the system, both before and after the must equal zero. Let's expand out the expressions for the final momentums of each car...

0 = p_1f + p_2f

0 = (m_car*v_1f) + (m_car*v_2f)

Divide both side by the mass of the cars...

0 = v_1f + v_2f

v_1f = - v_2f

So, conservation of momentum tells us that the final velocities of each car must be equal and opposite. But, this equation doesn't actually give us enough information to calculate what those equal and opposite velocities would be. At a lower bound, the final speeds of the cars would be zero, and the cars would "stick together". At an upper bound, the final speeds of the cars would be equal and opposite to their starting velocities -- but this is not realistic since it implies an elastic, energy-conserving collision, which a car crash definitely isn't.

In short, we don't quite have enough information to solve for their actual final velocities, but we know they should be equal and opposite to each other!

--Zach