How fast will the pin move immediately after it is hit?

Question

The rolling bowling ball strikes a pin (m = 1kg), sending it flying. How fast will the pin move immediately after it is hit? (Think: what equation should you use?)

 

Show work please.

Answer 1

The answer depends on the mass of the bowling ball and on how elastic the collision actually is.

If the the collision is assumed to be an elastic head-on collision, both energy and momentum are conserved.   Head- on collisions are inherently one dimensional.   For these, the energy conservation equation simplifies to:

 

    v1i + v1f = v2i + v2f    where v1 is the velocity of the bowling ball and v2 is the velocity of the pin.   The i and f refer to initial and final.   Not all textbooks cover this formula because it is limited to the one dimensional case.

 

  The conservation of momentum equation is more familiar:

 

    m1 v1i + m2 v2i =  m1 vif + m2 v2f.

 

  In both these equations  v2i = 0  and the desired  quantity is  v2f.   

 

There are two equations but three unknowns:  v1i (initial speed of bowling ball) ,  v1f (speed of bowling ball after collision),   v2f (the desired speed of pin after the collision.).    The standard approach is to rewrite the equations in terms of the ratios   v1f/v1i   and  v2f/v1i.    When this is done the ratio v1f/v1i can be eliminated leading to:

 

   v2f/v1i  =   2/[1 + m2/m1  ]

 

   If the bowling ball mass is 7 kg,   v2f/v1i =   1.75   

 

  The pin flies away with almost twice the precollision speed of the bowling ball. 

( The speed of the bowling ball after the collision is reduced to .75 times its precollision speed.  )

 

A technical note is that the simplified energy equation above depends on the conservation of momentum, but is independent of it.

  

Answer 2

The question does not give enough information to actually solve the problem, but the physics is easy. This is a conservation of momentum question. The initial momentum of the ball = m_B x v_{i (the subscript means the initial velocity of the ball).} Momentum is conserved in any collision, so the final momentum must equal the initial momentum. The final momentum = the momentum of the pin = m_P x v plus the momentum of the ball = m_B x v. If you put all those equations together,

m_B x v_i = m_P x v_f + m_B x v_{f ( the subscript f indicates the final velocity)}