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p0330 - 3.30 CHAPTER 3 PROBLEM 30 247 3.30 Chapter 3...

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3.30. CHAPTER 3, PROBLEM 30 247 3.30 Chapter 3, Problem 30 Problem: A particle of mass m is moving with constant velocity v = V i when it suddenly explodes. After the explosion, there are two fragments. One fragment moves at an angle 2 θ to the horizontal as shown with speed 3 2 V . Its mass is λ M , where λ < 1 is a constant. The other fragment of mass (1 λ ) M , moves at an angle θ to the horizontal as shown, also with speed 3 2 V . Solve for λ and θ . M v = V i λ M (1 λ ) M 2 θ θ 3 2 V 3 2 V . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Solution: The explosion gives rise to internal forces that cause the particle fragments to move away from the original path along which the particle moved. Since there are no external applied forces, momentum is conserved, wherefore MV i = 3 2 λ MV (cos 2 θ i + sin 2 θ j ) + 3 2 ( λ 1) MV (cos θ i sin θ j ) So, in terms of x and y components, we have MV = 3 2 λ MV cos 2 θ + 3 2 (1 λ ) MV cos θ 0 = 3 2 λ MV sin 2 θ 3 2 (1 λ ) MV sin θ Using the fact that sin 2 θ = 2 sin θ cos θ , the second equation becomes 0 = 3 λ MV sin θ cos θ 3 2 (1 λ ) MV
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