Direction as east and the positive y direction as

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-direction as east, and the positive y -direction as north. Over the time interval of 1.5 s during which they join hands, Allison exerts an average force on Bob of F A on B = (100 ˆ i + 200 ˆ j ) N (a) What are the x and y -components of the velocity with which the two skate o together? (b) What are Allison’s initial speed and direction before she joins Bob? (c) Let K denote the amount of kinetic energy lost when Bob and Allison come together. Suppose Allison has the same initial speed found in part (b), but her initial path is instead directly north. When they come together, would more or less kinetic energy be lost than the amount K ? Explain your answer. You don’t need to do any numerical calculations. (a) From the definition of average force, we have p Bf - p Bi Δ t = F AB p Bf = p Bi + F AB Δ t p Bf,x = p Bi,x + F AB,x Δ t = m B v B cos B + F AB,x Δ t = (70)(9) cos(30) + (100)(1 . 5) = 696 kg m / s p Bf,y = p Bi,y + F AB,y Δ t = - m B v B sin B + F AB,y Δ t = ( - 70)(9) sin(30) + (200)(1 . 5) = - 15 kg m / s v fx = p Bf,x m B = 696 70 = 9 . 94 m / s v fy = p Bf,y m B = - 15 70 = - 0 . 214 m / s (b) We can use conservation of momentum to determine Allison’s initial velocity. In the x -direction, we have the following: p ix = p fx m A v Ax + m B v B cos B = ( m A + m B ) v fx v Ax = ( m A + m B ) v fx - m B v B cos B m A = (50 + 70)(9 . 94) - (70)(9) cos(30) 50 = 12 . 9 m / s 2
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In the y -direction, we have the following: p iy = p fy m A v Ay - m B v B sin B = ( m A + m B ) v fy v Ay = ( m A + m B ) v fy + m B v B sin B m A = (50 + 70)( - 0 . 214) + (70)(9) sin(30) 50 = 5 . 79 m / s Allison’s initial speed and direction are therefore as follows: v A = q v 2 Ax + v 2 Ay = p 12 . 9 2 + 5 . 79 2 = 14 . 2 m / s A = arctan v Ay v Ax = arctan 5 . 79 12 . 9 = 24 . 1 deg (c) Allison is initially traveling 24.1 deg north of east. This makes an angle of 54.1 deg with Bob’s initial trajectory. If Allison were initially traveling north, then her trajectory would make an angle of 120 deg with Bob’s initial path. Therefore, upon colliding, Bob and Allison would each undergo a larger change in direction, change in momentum, and therefore loss of kinetic energy. The kinetic energy lost would therefore be larger than the amount K lost in the collision described in the problem.
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