# Elastic Collisions - Equation not working?

This is the question:

When cars are equipped with flexible bumpers, they will bounce off each other during low-speed collisions, thus causing less damage. In one such accident, a 1850 kg car traveling to the right at 1.60 m/s collides with a 1450 kg car going to the left at 1.10 m/s . Measurements show that the heavier car's speed just after the collision was 0.270 m/s in its original direction. You can ignore any road friction during the collision.

I solved this problem correctly with conservation of momentum: (m1v1 + m2v2)i = (m1v1 + m2v2)f. The result was 0.597 m/s.

However, when I attempted to use the equations for elastic collisions, shown here:

I ended up with v2,f = 1.927 m/s. Why is the second equation in the picture wrong? I thought that if a collision is elastic, these equations always apply.

• They may cause less damage to the cars, but not necessarily to the occupants. In an inelastic collision the vehicles crumble absorbing energy that would otherwise be absorbed by the occupants. Nov 14, 2019 at 23:57
• We have MathJax running on the site which means that you can write mathematics that will be neatly marked up using more or less LaTeX math-mode syntax. This is prefered to weither plain ascii math or (especially) images of math. Nov 15, 2019 at 1:14
• Follow the procedure I outlined in this answer to check your work. Nov 16, 2019 at 15:36
• If this collision conserved energy, a final velocity would not be given. It could be calculated. (I would call this a non-elastic collision.) Nov 16, 2019 at 18:42

The equations for plastic collision with a coefficient of restitution $$\epsilon$$ are
\begin{aligned} v_1^f & = \frac{m_1 - \epsilon m_2}{m_1+m_2}v_1 + \frac{\epsilon m_2+m_2}{m_1+m_2} v_2 \\ v_2^f & = \frac{\epsilon m_1+m_1}{m_1+m_2} v_1 + \frac{m_2 - \epsilon m_1}{m_1+m_2} v_2 \end{aligned}