- with constant acceleration a directed along the negative -axis. The equation of motion of the particle has the form where and are positive constants. Then the velocity of the particle at the origin is?
A particle moves in a the place
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The general equations for motion with constant acceleration are { x = x 0 + v x , 0 t + 2 1 a x t 2 y = y 0 + v y , 0 t + 2 1 a y t 2 In this case we have x 0 = y 0 = 0 , a x = 0 , a y = − a , so that { x = v x , 0 t y = v y , 0 t − 2 1 a t 2 Write t = x / v x , 0 to eliminate t in the second equation: y = v x , 0 v y , 0 x − 2 v x , 0 2 a x 2 . Compare with the given form y = p x − q x 2 to conclude p = v x , 0 v y , 0 , q = 2 v x , 0 2 a . Solve for the initial velocity components: { v x , 0 = 2 q a ; v y , 0 = p v x , 0 . The magnitude of the initial velocity is then v 0 = v x , 0 2 + v y , 0 2 = ( 1 + p 2 ) v x , 0 2 = ( 1 + p 2 ) 2 q a = 2 q a ( 1 + p 2 2 ) .