James1019 said:
Can you give me the answer or the equation?
It is a little bit confusing
I am only a student and not an expert, but projectile motion is the subject I master the most and am most familiar with.
As you know, this type of motion describes a parabola, with a composition of two motions: rectilinear along the horizontal axis, uniformly accelerated rectilinear (with acceleration equal to ##g##) along the vertical axis.
One way to solve this problem is to find the analytical expression of the equation of the maximum height ##y_{\text{max}}## as a function of the initial velocity (and, in particular, of the latter's vertical component). As
@haruspex pointed out well, what happens to the velocity of the projectile when the body reaches its maximum height, i.e. the vertex of the parabola? Once you have succeeded in deduce it, you can calculate (as a function of the initial vertical velocity) the time it takes for the projectile to reach the maximum height. Substitute this time into the equation of motion of the body along the vertical axis (analogous to that of a body thrown upwards with vertical velocity and subject only to the downward acceleration of gravity, i.e. uniformly decelerated motion) and you will find the expression for the maximum altitude. From there, you can calculate the initial vertical velocity.
In addition, you are given the range, i.e. the maximum horizontal distance, from the text. Carry out a procedure similar to the previous one and you will obtain an expression as a function of the horizontal velocity (constant) and the initial vertical velocity. You have already found this velocity previously, replace it, and thus obtain the horizontal velocity.
There is no need for angles or trigonometric functions of any kind.
I agree with
@haruspex, it is much better to work in symbols than with numbers initially. If you can, substitute numbers only at the end.