If you suppose that at the time the speed along the x axis is null [unfortunately that isn't...] the problem is onedimensional and the 'move equation' is...

, , (1)

Setting the (1) becomes...

(2)

... that is more 'tractable' if you swap the u with the t so that...

(3)

Integrating both terms of (3) you obtain...

(4)

... and from it...

(5)

Integrating both terms of (5) and taking into account that is...

(6)

... we obtain finally...

(7)

At this point you have to solve two distinct problems...

a) first you solve the (1) for with and initial conditions , ...

b) second you solve (1) for with and initial conditions determined by the step a)...

Kind regards