  Question

A ball of mass $$5kg$$ is suspended from a chord of length $$l=0.5m$$ is attached to a cart $$A$$ of mass $$2.5kg$$, which slides freely on a frictionless horizontal track as shown in figure. The ball given an initial horizontal velocity $$1m/s$$ relative to the $$ground$$ while the cart is at rest.Find(a) The velocity of the ball when string makes $$minimum$$ angle with vertical(b) The maximum vertical distance through which the ball will rise. Solution

(a) At time of horizontal velocity of ball is same of cart velocity .By linear momentum conservation $$5x_{ 1 }=5xv+2.5v\\ \cfrac { 5 }{ 7.5 } =v\\ v=\cfrac { 2 }{ 3 } m/s$$(b) By energy conservation theorem $$\cfrac { 1 }{ 2 } 5(1)^{ 2 }=\cfrac { 1 }{ 2 } 5\times { v }^{ 2 }+\cfrac { 1 }{ 2 } 2.5\times { v }^{ 2 }+5g(h)\\ 5=\cfrac { 20 }{ 9 } +\cfrac { 10 }{ 9 } +10gh\\ 5-\cfrac { 30 }{ 9 } =10\times 10h\\ \cfrac { 15 }{ 9 } =100h\\ h=\cfrac { 5 }{ 300 } m\\ =\cfrac { 5 }{ 3 } cm$$ Physics

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