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A particle of mass m is fired vertical...

A particle of mass m is fired vertically upward with speed `v_(0) (v_(0) lt " escape speed ")` . Prove that
(i) Maximum height attained by the particle is `H = (v_(0)^(2)R)/(2gR - v_(0)^(2))` , where g is the acceleration due to gravity at the Earth's surface and R is the Earth's radius .
(ii) When the particel is at height h , the increase in gravitational potential energy is `(mgh)/(1+h/R)`

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To solve the problem step by step, we will use the principles of conservation of energy and gravitational potential energy. ### Part (i): Maximum Height Attained by the Particle 1. **Initial Energy Calculation**: The initial kinetic energy (KE) of the particle when it is fired is given by: \[ KE_i = \frac{1}{2} mv_0^2 ...
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