Consider strictly one-dimensional head-on collisions of particles, as shown below. m?
has an initial velocity v?? while m? is initially at rest. During the collision, some of m?'s
kinetic energy is transferred to m?.
a) Let us first assume that all collisions are fully elastic. In lecture it was suggested that, if m?
?m?, the kinetic energy transfer from m? to m? is not perfect but can be enhanced by placing a
third mass, m?, also initially at rest, between m? and m?. In this case m? first collides with m?,
after which m? alone collides with m?. Show that the value of m? that maximizes $k_{2f}/k_{1i}$ is m?
= $(m_1m_2)^{1/2}$. Here $k_{2f}$ is the kinetic energy of m? after the collision, and $k_{1i}$ is the kinetic energy
of m? before the collision.
For parts b) and c) we now assume always fully inelastic collisions for all particles.
b) m? is absent. m? (moving with velocity v??) collides with m? (initially at rest). Determine
the ratio of kinetic energy of the system after the collision to kinetic energy in the system
before the collision, $K_f/K_i$. Note that this energy ratio is defined differently from that in a).
c) Following our strategy in a), we insert a third particle, m?, again initially at rest, between
m? and m?. Now m? first collides with m?, and then there is a collision with m?. Determine if
the insertion of m? can increase $K_f/K_i$ for the system (where again $K_i$ is the kinetic energy of
the system before any collisions and $K_f$ is the kinetic energy of the system after all collisions)
above that calculated in b). If m? can enhance the kinetic energy ratio, what value of m? (in
terms of m? and m?) maximizes the ratio?
