A ball thrown at an angle has exactly one force on it, gravity. Air resistance is left out. And that gravity points straight down the whole time it is in the air, with an unchanging size.
The ball, however, moves at an angle. The direction of the motion and the direction of the force disagree, which makes the problem awkward as it stands. So we split something in two, and what we split is not the force but the velocity: one part horizontal, one part vertical.
Write the initial speed as and the angle above the horizontal as . The horizontal component is and the vertical component is . The rectangle in the figure shows that adding those two returns the original initial velocity.
From here on the two are followed separately. They are not split up only to be recombined at the end: they may be kept apart the whole way through. The next two scenes show why that is allowed.
Take the horizontal direction first. No force acts along it. Gravity points straight down, so it has no horizontal component whatsoever.
With no force there is no change in velocity. The horizontal speed stays at and never changes once between launch and landing. This is uniform motion in a straight line, plain and simple.
The distance covered is . The marks laid along the ground in the figure are where the shadow stands at equal intervals of time. They are evenly spaced, and the ball's shadow crosses them in strides of equal length.
The ball climbs and then descends, yet the shadow never alters its pace. The up and down motion has no effect at all on how the ball advances sideways.
Now the vertical direction, which takes the whole of gravity. With upward as positive, the velocity runs as and the height as .
Those two equations should look familiar. Letter for letter they are the equations of a ball thrown straight up at a speed of . The vertical motion of a ball flying at an angle was a throw upward all along.
On the right of the figure a ball thrown straight up runs alongside. A dashed line joins it to the ball flying at an angle, and that line stays horizontal for as long as you care to watch. The two rise together, reach the top together and come down together.
The marks running up the side are the heights at equal intervals of time. They crowd together near the top because the ball is slowing as it climbs. At the top reaches zero and the motion turns into a descent. Compare them with the evenly spaced marks along the ground.
One experiment shows better than any other that the horizontal and the vertical may be treated separately. From the same height , one ball is thrown horizontally and the other is simply released.
The ball thrown horizontally has an initial vertical velocity of zero. So does the ball that was released. And in that direction gravity is the only force on either of them. Same height, same initial velocity, same force. So both follow and come down in exactly the same way.
The two balls in the figure are level with each other at every moment, and they reach the ground together. The time to fall is , and the horizontal speed does not appear in it. However hard the first ball is thrown, the falling takes just as long.
The sideways speed has nothing to do with the falling. That is what it means to say the horizontal and the vertical move separately. It is not that we are permitted to consider them apart. They were two separate motions laid on top of each other from the start.