Based on the given information, we can write the governing equation in this form

    h(t) = .


In this equation, coefficient 5 at    is half of the gravity acceleration,
which is 10 m/s^2  (or, more precisely, we take this value of 10 m/s^2 for the gravity
acceleration as an approximation, instead of its true value of 9.81 m/s^2).


Next, the constant term of 50 in this equation represents the initial height of 50 m.


Coefficient "v" is the only value, which we do not know now.


This value is the starting velocity.


        There is one more agreement, which is commonly accepted in this class
        of problems: the coordinate axis for the height is directed vertically up.
        It dictates the signs of the terms with    and  with "t".


To find the starting velocity, we can use the given condition about reaching the maximum height.


The theory says that the maximum height is achieved at  t = , referring to coefficients
of the general form equation.


In your case a = -5,   = 2, and it gives you this equation for v

     = 2,

or

     = 2,

    v  = 2*10 = 20.


Thus, the initial vertical velocity is 20 m/s, and your governing equation for the height is 

    h(t) =  + 20t + 50.      ANSWER

    Moreover, the given value 75 m for the maximum height is INCORRECT 
        and is NOT CONSISTENT with the given time of 2 seconds : 
            it is only PURE FANTASY of a problem's composer.

    The correct value for the maximum height is 70 m in this problem,
    and it CAN NOT be different - otherwise it is not consistent 
    with the Physics laws.


    The time "6 seconds" in the problem is incorrect and inconsistent with 
    2 seconds given to reach the maximum height.


    My DIAGNOSIS after writing my solution is that the problem, 
                 as it is presented in the post,
        is composed by a person UNFAMILIAR with the subject.


    AS I SEE IT SEVERAL TIMES per day EACH and EVERY DAY at this forum.


    I tried to teach you the truth as much as I could.