Particle functions

Question

Consider a particle moving along the x-axis where x (t ) is the position of the particle at time t, x' (t ) is its velocity, and x'' (t ) is its acceleration. 

A particle moves along the x-axis at a velocity of v ( t ) = 3/√t , t > 0. At time t = 1, its position is x = 11. Find the acceleration and position functions for the particle. 

Answer 1

a(t) = dv/dt = (-1/2)(3t^-3/2) = (-3/2)t^-3/2

x(t) = ∫v(t)dt = ∫3t^-1/2dt = 3t^1/2/(1/2) + c = 6t^1/2 + c

x(1) = 11

11 = 6(1)^1/2 + c

11 = 6 + c

c = 5

x(t) = 6t^1/2 + 5