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241_PartUniversity Physics Solution

241_PartUniversity Physics Solution - 7-30 Chapter 7(e x1 =...

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7-30 Chapter 7 (e) 1 0 3 x x = , and 0 2 0 2 (3 ) 9 U x x α = − . 2 2 0 0 0 0 1 2 2 2 0 0 0 2 2 2 2 ( ) ( ( ) ( )) 2 9 9 x x x x v x U x U x x x mx m m x x x x α α α = = = . The particle is confined to the region where 1 ( ) ( ) U x U x < . The maximum speed still occurs at 0 2 x x = , but now the particle will oscillate between 1 x and some minimum value (see part (f)). (f) Note that 1 ( ) ( ) U x U x can be written as 2 0 0 0 0 2 2 0 0 2 1 2 , 9 3 3 x x x x x x x x x x α α ⎤ ⎡ + = ⎥ ⎢ ⎦ ⎣ which is zero (and hence the kinetic energy is zero) at 0 1 3 x x x = = and 3 0 2 x x = . Thus, when the particle is released from 0 x , it goes on to infinity, and doesn°t reach any maximum distance. When released from 1 x , it oscillates between 3 0 2 x and 0 3 x . E VALUATE : In each case the proton is released from rest and ( ) i E U x = , where i x is the point where it is released. When 0 i x x = the total energy is zero. When 1 i x x = the total energy is negative. ( ) 0 U x as x → ∞ , so for this case the proton can’t reach x → ∞ and the maximum x it can have is limited. Figure 7.87

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8-1 M OMENTUM , I MPULSE , AND C OLLISIONS 8.1. I DENTIFY and S ET U P : . p mv = 2 1 2 . K mv = E XECUTE : (a) 5 (10,000 kg)(12.0 m/s) 1.20 10 kg m/s p = = × (b) (i) 5 1.20 10 kg m/s 60.0 m/s 2000 kg p v m × = = = . (ii) 2 2 1 1 T T SUV SUV 2 2 m v m v = , so T SUV T SUV 10,000 kg (12.0 m/s) 26.8 m/s 2000 kg m v v m = = = E VALUATE : The SUV must have less speed to have the same kinetic energy as the truck than to have the same momentum as the truck. 8.2. I DENTIFY : Example 8.1 shows that the two iceboats have the same kinetic energy at the finish line. 2 1 2 K mv = . p mv = . S ET U P : Let A be the iceboat with mass m and let B be the iceboat with mass 2m , so 2 B A m m = . E XECUTE : A B K K = gives 2 2 1 1 2 2 A B mv mv = . 2 B A B B A m v v v m = = . A A A p m v = . ( ) (2 ) / 2 2 2 B B B A A A A A p m v m v m v p = = = = . E VALUATE : The more massive boat must have less speed but greater momentum than the other boat in order to have the same kinetic energy. 8.3. I DENTIFY and S ET U P : p mv = . 2 1 2 K mv = . E XECUTE : (a) p v m = and 2 2 1 2 2 p p K m m m = = . (b) c b K K = and the result from part (a) gives 2 2 c b c b 2 2 p p m m = . b b c c c c 0.145 kg 1.90 0.040 kg m p p p p m = = = . The baseball has the greater magnitude of momentum. c b / 0.526 p p = . (c) 2 2 p mK = so m w p p = gives m m w w 2 2 m K m K = . w mg = , so m m w w w K w K = . m w m m m w 700 N 1.56 450 N w K K K K w = = = . The woman has greater kinetic energy. m w / 0.641 K K = . E VALUATE : For equal kinetic energy, the more massive object has the greater momentum. For equal momenta, the less massive object has the greater kinetic energy.
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241_PartUniversity Physics Solution - 7-30 Chapter 7(e x1 =...

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