Kinetics Problem: Head-On Collision Between Two Cars

[Graador]

Homework Statement

If two cars of equal mass hit each other head on, car A travelling at 25 m/s towards the right and car B travelling at 20 m/s to the left, what will be the speed of each car post-collision if 75% of the energy is lost in bending the metal of the cars.

Relevant Equations

m1v1 + m2v2 = m1V1 + m2V2
e = Kf/Ki

Since the mass of both vehicles is the same, it's possible to calculate Ki which happens to be 512,5J and from there, multiply it by 0,25 since 75% of the energy is gone and I end up with 128,125J.

Now my problem is that for the velocity, I have: 25,0 + -20,0 = V1 + V2 which is 5,00 = V1 + V2

When I take Kf, I have: 128,125 = V1(sq)/2 + V2(sq)/2 so 256,25 = V1(sq) + V2(sq)
If I try and get the square root, I end up with 16,0 = V1 + V2 which doesn't fit with the first equation which said 5,00 = V1 + V2

After some trial and error, I found that the velocity of the first car is -8,5 m/s and the second 13,5 m/s which when added up, equal 5,00 and when their squares are added up and divided by two, equal 127,25 which is close enough. I just don't understand how to find both velocities from the actual equations themselves because all I find is a quadratic formula that gives me wrong answers.

 

[kuruman]

You will have to solve a system of two equations and two unknowns. This is best done when you first write down equations using symbols, then substitute the numbers after solving the system symbolically. Your energy balance equation seems incorrect.

 

[Graador]

You will have to solve a system of two equations and two unknowns. This is best done when you first write down equations using symbols, then substitute the numbers. Your energy balance equation seems incorrect.

So if my energy equation is correct and my first equation is also correct (hard to calculate that one wrong), what am I supposed to do? Is there an equation that I have not considered? I already tried swapping things from on formula to another but they don't equal the same thing for V1 + V2 so that didn't work.

 

[kuruman]

Write down two equations using symbols for the the different quantities that you have. Use different symbols for different quantities. The first equation is momentum conservation, the second equation is the energy balance equation which says that the initial kinetic energy of the cars is equal to the final kinetic energy of the cars plus the energy loss in bending metals. The two unknowns are the final velocities of the cars.

 

[haruspex]

755 of the energy is lost in bending
...
calculate Kf which happens to be 512,5J
...
256,25 = V1(sq) + V2(sq)
If I try and get the square root, I end up with 16,0 = V1 + V2

I take it that should be 75%.
...
You mean Ki, and since you do not know the mass it is 512,5m ms^-2, where m is the mass of each car.
...
$\sqrt{x^2+y^2}\neq x+y$ (unless one or both is zero).

 

[PeroK]

So if my energy equation is correct and my first equation is also correct (hard to calculate that one wrong), what am I supposed to do? Is there an equation that I have not considered? I already tried swapping things from on formula to another but they don't equal the same thing for V1 + V2 so that didn't work.

This may be simpler in the centre of momentum frame, where you have a symmetry of the motion.