A particle is projected vertically upwards with a velocity of 196 m/sec. How much will be the greatest height?

A particle is projected vertically upwards with a velocity of 196 m/sec. How much will be the greatest height? Correct Answer 1960 m

Let the particle projected from A with velocity u = 196 m/sec and P be its position at time t where, AP = x m, if v m/sec be its velocity at P,then the equation of motion of the particle is, dv/dt = -g ……….(1) Since v = u, when t = 0, hence from (1) we get, u∫vdv = -g 0∫tdt Or v – u = -gt Or dx/dt = u – gt ……….(2) Since x = 0, when t = 0, hence from (2) we get, 0∫x dx = 0∫x (u – gt)dt Or x = ut – (1/2)gt2 ……….(3) Again, (1) can be written as, dv/dx*dx/dt = -g Or v(dv/dx) = -g ……….(4) Since v = u, when x = 0, hence, from (4) we get, u∫vvdv = -g 0∫xdx Or v2 = u2 – 2gx ……….(5) If H m be the greatest height of the particle, then v = 0, when x = H; hence, from (5) we get, 0 = u2 – 2gH Or H = u2/2g = (196*196)/(2*9.8)   = 1960 m.