Moving car hits a stationary car

[physicsman2]

Homework Statement

If a moving car hits a stationary car and they become attached how would the velocity change for both cars

Homework Equations

3. The Attempt at a Solution [/b
I believe that the moving cars velocity will decrease and the stationary cars velocity will increase so that the momentum of both cars is equal

 

[AEM]

I suspect you are making the assumption that the masses of the cars are equal. Conservation of momentum holds and you can derive an expression that will give you the tangled mess in terms of the initial velocity(s) and masses.

 

[Galadirith]

Hi physicsman2, well you are pretty much right, you correct identified that the velocity of the moving car would decrease and the the velocity of the stationary car would increase, however you second comment you made about them having the same momentum isn't necessarily correct as consider each car may have a different mass. If they had different masses but had equal momentum then their velocities would not be equal, but that obviously can't be true as they are attached to one another, so they must have the same velocity, and that is the distinction you need to make clear to yourself, there velocities would be the same but not necessarily their momentum if their masses are not equal.

So that's a qualitative answer that you gave, but it might be easier to approach it from a mathematical perspective. So let's first set up our scenario, we have car 1 of mass m₁, which is traveling with initial velocity u₁ . We then have a stationary car, well call car 2 of mass m₂, which may or may not be the same as the mass of car 1, and car 2 has initial velocity u₂ which as it is stationary equals 0. so let us put this all into an equation which describes how momentum is conserved. I do apologise if you haven't learn't this concept yet. So the equation describing the conservation of momentum:

m_1 u_1 + m_2 u_2 = m_1 v_1 + m_2 v_2

where v₁ is the final velocity of car 1 and v₂ is the final velocity of car 2. now v₁ = v₂ as we established earlier, so we will call this common final velocity v. so inputing this and our previous properties into the equation we get:

m_1 u_1 + m_2 (0) = m_1 v + m_2 v
m_1 u_1 = m_1 v + m_2 v

now what is already evident is that the speed of car 2 has increase, but can we show from this equation the the speed of car 1 has decreased, well we have to be able to show that v < u₁, so:

m_1 u_1 = m_1 v+ m_2 v
u_1 = v\left( \frac{m_1 + m_2}{m_1}\right)

now hopefully it should be clear that the fraction involving the masses must be greater than one as all the masses are positive, this therefore must mean that v < u_1. :D

I hope that helped, you initial answer was correct, but you simply made assumptions about the situation which were not necessarily correct :D