What are the equations for solving the collision problem in momentum?

[Violagirl]

Homework Statement

A car moving to the right at 40 m/s has a mass of 2000 kg. It strikes a car of mass 1500
kg and both cars immediately come to rest. What was the speed and direction of the
second car just prior to the collision?

Homework Equations

p_initial=p_final

p=mv

K=1/2mv²

The Attempt at a Solution

In determining the speed of the second car, I set up my formula to try to resemble what the problem provides in how the first car is moving:

If p_initial=p_final, then:

m₁v₁=m₂v₂

I set it up like this since we're only told that the first car is traveling in a given direction towards the second car and we we want it so that we can find v₂. So:

(2000 g) (40 m/s) = (1500 g)v₂

v₂=53.3 m/s

Otherwise, I'm not sure what approach to take the find the direction. I thought maybe in the x direction with sin equaling 180 degrees with car one in applying the kinetic energy equation but I am not exactly sure if this is correct.

 

[Simon Bridge]

If p_initial=p_final, then:

m₁v₁=m₂v₂

What does that imply about the direction of the second car?

I'm not sure what approach to take the find the direction. I thought maybe in the x direction with sin equaling 180 degrees with car one in applying the kinetic energy equation but I am not exactly sure if this is correct.

If "to the right" (the direction of the 1st car) is the +x direction, then 180 degrees from the +x direction is what?

You could try doing it formally - call the direction of the first car +x and write out vector equations:

before collision:
$\vec{p}_{b}=p_1\hat{\imath}+\vec{p}_{2}=(m_1v_1+m_2v_2\cos{\theta})\hat{\imath}+m_2v_2\sin(\theta)\hat{\jmath}$

after collision:
$\vec{p}_{a}=0$

conservation of momentum:
$\vec{p}_a = \vec{p}_b$

... gives you two equations and two unknowns.

If $\theta=0$ then car-2 is traveling in the same direction as car-1; which is to say: "to the right".
To get full marks, you should always try to provide final answers using the same terms as used in the question.

In general: if you lay out all conservation-of-stuff problems like this, you'll find them easier to think your way through.

Note: write the expression for total kinetic energy before and after the collisions.
Is there any value for $v_2$ (your only unknown) which will make KE before equal KE after? i.e. is kinetic energy conserved in the collision?

 

[Basic_Physics]

Both cars are moving initially and "finally" both are stationary.
So p_initial need to have both components and p_final = ...