BOBON4dHHHHHHYxHY$! N,-Zx HHHV,(hh hd'0F d .xS 0l `/kxSL` 0O@`0zN `PixStL zN `PixSL "L2kh 1H a4 N4(}0DSET /<Y\YhvYdYl v }v~     Q ]    ' Q    , -            2     7 J S g    + N o } Q S       ( ) C D f g     S X          " # % & S X \ b f k u v x y               , N P         d e                                " # ( + , 0 1 2 R S                 : ; ? @ B C H I M N W X \ ] _ ` e f          !  "  R  S  W  X  Z  [  `  a  m  n  s  t          X  Y  ^  _  b  l  Q  S     !    !    !      "   $ " .       !    !    !    !          #                  !   !     o p v w ) $. %5 $c   __ >bb && ? QWKK(M|iY` O$More on the Ballistic Pendulum The speed of the dart can be calculated twicefrom the recoil of the gun and also from the collision with the block. Both the gun and the block are suspended as ballistic pendula (pendulums). Throughout; m is the mass of the dart, v is the speed of the dart, M1 is the mass of the gun, V1 is the recoil speed of the gun, M2 is the mass of the block, and V2 is the speed of the block and dart after the dart hits and sticks to the block. First, look at the recoil of the gun:  The gun and dart are initially at rest so their initial total momentum is zero, Ptot,i = 0 After the dart leaves, the final total momentum must still be zero. The dart carries momentum to the right of pdart,f = m v while the gun carries momentum to the left of pgun,f = M1 ( V1) = M1 V1 The final total momentum is still zero so Ptot,f = pdart,f + pgun,f = m v M1 V1 = 0 = Ptot,i m v = M1 V1 v = [M1 / m ] V1 That is, once we can determine V1, the recoil speed of the gun, we can use this equation to determine v the speed of the dart. Now we turn our attention to the inelastic collision of the dart hitting and sticking onto the block:  The initial total momentum is just the momentum of the incoming dart, pblock,i = 0 pdart,i = m v Ptot,i = m v Finally, after the collision, the dart and block move off together, with a common speed of V2. Therefore, the final total momentum is Ptot,f = (M2 + m) V2 Since momentum is conserved, this means Ptot,f = (M2 + m) V2 = m v = Ptot,i m v = (M2 + m) V2 v =[ (M2 + m) / m ] V2 That is, once we can determine V2, the common speed of the block and dart after their inelastic collision, we can use this equation to determine v the speed of the dart. Now, how can we determine V1 or V2? M1 and M2 are each suspended as a ballistic pendulum. If a ballistic pendulum of mass M has an initial speed of V, how can we measure V in terms of the vertical distance h which the pendulum rises? We can use conservation of energy for that.  Initially, all of the energy is in the form of Kinetic Energy, KEi = (1/2) M V2 PEi = 0 Ei = (1/2) M V2 At the end of the swing, the mass stops momentarily. This means the final Kinetic Energy is zero and all the energy is now in gravitaional Potential Energy, KEf = 0 PEf = M g h Ef = M g h Using Energy Conservation, we have Ei = (1/2) M V2 = M g h = Ef V2 = 2 g h V =  So far, everything should have been very straightforward. We have used momentum conservation to transform the problem from measuring a high velocity like v to measuring a slower velocity V (V can be either V1 or V2). Now we have used energy conservation to transform the problem from measuring a velocity V to measuring a vertical distance h. However, it is still very difficult to measure this vertical distance h. Rather than attempting to do that, we will measure the horizontal distance d, the horizontal distance that the ballistic pendulum swings from its position at rest to its maximum swing. We will measure this with a small, lightweight paper rider on the meter stick below the ballistic pendulum.  From this diagram, we can use the Pythagorean Theorem to write l 2 = (l h)2 + d2 where l is the vertical length of the strings supporting the pendulum, h is the small vertical distance we need, and d is the horizontal distance we can measure rather easily. Then we can solve for h in terms of l and d, l 2 = l 2 2 l h + h2 + d2 But h is very small so h2 is very small so we can safely omit it (h2 << d2, h2 << l h), 2 l h = d2 h = d2 / 2l We can easily measure l and d. From these we can determine h. From h we can determine V (or V1 and V2 if you prefer). From V we can determine v, the speed of the dart. Find the darts speed by measurements on the gun. Then find the darts speed by measurements on the block. Then compare these two values of v, the speed of the dart (that is, find their percent difference). What does all this mean? What ideas or principles have you used? 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