Three body problem + spring + center of mass (classical mechanics)

Question

Given three bodies with mass $m$ each - body $1$ moves (to the right) towards the stationary body $2$ with velocity $v_1=7v$ and collide with it. On the other side of body $2$, body $3$ moves (to the right) with velocity $v_3 = v$, with a massless spring (with a constant $k$) attached to its left part. After the collision of bodies $1,2$, they move together towards body $3$, and contract the spring as they collide with it. What is the maximal contraction of the spring?

To calculate this, I started by calculating the velocity of the center of mass, which is $v_{cm}=frac{8}{3}v$ (this is correct). Afterwards, I tried using conservation of energy, which did not result in the right answer (I do not know what the right answer is, just that the answers I submitted). My attempts are:

Equating the energy at the time of maximal contraction and energy in the beginning of the experiment.
$$
frac{1}{2} (3m) cdot v_{cm}^2 + frac{1}{2} k (Delta x)^2 = frac{1}{2} m v_1^2 + frac{1}{2}m v_3^2 
$$

Equating the energy at the time of maximal contraction and energy after the collision of bodies $1$ and $2$ .
$$
frac{1}{2} (3m) cdot v_{cm}^2 + frac{1}{2} k (Delta x)^2 = frac{1}{2} (2m) cdot left( frac{1}{2} v_1 right) ^2 
$$

What am I getting wrong here?

user256872 · Accepted Answer

You can't use conservation of energy for the first collision. The collision is inelastic, so kinetic energy will not be conserved (because the objects stick together). You must therefore use conservation of momentum to solve for the velocity of objects 1 & 2 after their collision. (energy is conserved, but it's transformed into other forms of energy that you cannot calculate given your data)
Then, for the collision between (1 & 2) & 3, you can use conservation of energy to figure out how the spring compresses.
You do not really need the CoM for this problem (unless you want to work with CoM frame for the first collision, but that's quite unnecessary).

Notwen · Answer

You are missing a term while writing the energy of system after 2nd colision that is:
$$
frac{1}{2}{mu}v_{rel}^2
$$
The final energy is
$$
frac{1}{2}{m}v_{cm}^2 + frac{1}{2}{mu}v_{rel}^2
$$
where $mu$ is the reduced mass: $frac{m_1m_2}{m_1 + m_2}$
EDIT: I wrote the equations for 2nd collision to simplify the calculation, you can work it out in ground frame but in my opinion it is better to use concept of reduced mass which reduces some calculation part.

Three body problem + spring + center of mass (classical mechanics)

2 Answers

Add your own answers!

Ask a Question