Why are x and y equal?

Question

This is the question I am doing:

A small object is projected with speed 24 m/s from point O at the foot of a plane inclined 45 degrees to the horizontal. The angle of projection is 15 degrees above the plane. After time t s, horizontal and vertical displacement are x m and y m.

Express x and y in terms of t, and hence find the value of t for the instant when the object strikes the plane.

So far I have expressed x and y correctly by the mark scheme-

x = (24cos60)t

y = (24sin60)t - 5t^2

But going by the mark scheme, it says the next step is to set x and y equal to each other and solve for t. How do I know that x and y would be equal at this point?

Answer 1

Drawing the described plane, a 45 deg inclination from the horizontal, onto an xy-coordinate system, this is equivalent to drawing the line y = x, with the x-axis representing 0 degrees and the y-axis representing 90 degrees from the horizontal.

Another way to think about it is if you visualize a literal enormous ramp with an inclination of 45 degrees above horizontal ground, any position along that ramp (remembering the special 45-45 right triangle that has equal base and height) is going to be equally as high as it is far horizontally.

In either model, along the plane, the x and y displacement will be equal.