Q5.8 Two spheres colliding (S)

Two titanium spheres approach each other head-on with the same speed and collide elastically. After the collision, one of the spheres, whose mass is 300 g remains at rest.
  1. What is the mass of the other sphere?
  2. Find the speed of the two-sphere center of mass both before and after the collision, if the initial speed of each sphere was 2 ms-1. Check each explicitly.
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Hint

Kinetic energy and momentum are conserved: the collision is elastic.
 

Solution

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Solution

Let’s draw a diagram and introduce some notation

Graphic - No title - twosphere

Let the two spheres have masses and m1 and m2 respectively. The initial velocities of particles 1 and 2 are v and -v; after the collision they are 0 and v2.

  1. To calculate the mass of the other sphere, let’s write down equations for conservation of energy and momentum:

    Momentum: 

    m1v-m2v=m2v2

    Kinetic energy: 

    \[ \frac{1}{2}m_1 v^2 + \frac{1}{2} m_2 v^2 = \frac{1}{2} m_2 v^2_2 \]

    We can easily solve these two equations to get

    m2=m1/3=100 g;

  2. The speed of the center of mass is a constant, since there is no external force acting on the system. We can calculate it any way we choose.

    Using Key Point 4.2 before the collision

    \[ v_{cm} = \frac{m_1 v - m_2 v}{(m_1+m_2)} = \frac{v}{2} = 1\,ms^{-1} \]

    Using Key Point 4.2 after the collision

    \[ v_{cm} = \frac{m_2 v_2}{(m_1+m_2)} = \frac{v}{2}\,\,{\rm also} \]