Ok, another rambling about swing science, physics, and physiology. I'm going to torture you guys some more since I have to read about the complaining about hitting formats, grid widths, and who has the fastest CHS and BS.
-why horizontal speed is converted to vertical speed and finally to club speed
Horizontal speed is converted to vertical speed in the take away and swing transition and does something important. The WHY part is to increase GFRs (ground force reactions) and SSCs (stretch shorten cycles) to the maximum to propel a large pulse of force through the kinetic chain of the golf swing.
A lot is mentioned about "tempo" by the golf instruction industry, but I do not know of anyone mentioning how it contributes to the physics of increasing the compound pendulum effect of the swing. Both increases in GFRs and SSCs can be observed if a wide take away with high hands is used to make the swing potentially more powerful. This post is about What is happening, HOW it is done and HOW you can do it too by learning what velocity has to do with timing. Imagine yourself addressing the ball and making your first moves.
If done optimally with the right speed the weight transition to the rear leg is "loaded". As the hands reach the top of the take away the speed causes a "lift" of the body and the club head goes past parallel in synchronicity with the "drop" of the body. Once the body drops it then enables the rear leg to drive forward and push toward the front foot and rotate the pelvis creating torque beyond what is normal. The "load-lift-drop-squat" sequence creates a powerful SSC that produces a high amount of GFRs that generate forces that start a powerful force through kinetic chain of the swing.
It should be noted that this additional swing force is being generated by the take away and this take away velocity is critical. Too slow and the load-lift-drop-squat is not producing a solid pulse of inertial force that can be transferred. It becomes fragmented and dissipates. if the take away velocity is too fast, the force disrupts balance and the direction of inertial forces can cancel or repel muscular forces. Finding the right speed for horizontal to vertical movements is dependent on the person and how they load the club lag during the take away and transition. Their weight and build/flexibility have something to do with it too so experiment.
if you are not sure what this feels like or do not understand this, go to the gym and do barbell curls until failure. Once you can no longer do strict curls you can then cheat by using your hips and legs. This "cheating" method in the gym is frowned upon, but is easy to do to lift the weight more times. Same thing with standing overhead presses. Since the weight is much heavier than a club, and is slower, it is easy to do naturally. Doing this same phenomenon with a LD golf swing is a bit more complex since the motion has more movements, is faster, and the club is very light. And that is why the speed of doing it is important and only the best have timed it right.
Most of the training info on how to hit a golf ball 400 yards is mostly about the swing and the muscles that drive the club into impact. Not much is focused on the take away and transition techniques that enable that swing to be as powerful as possible. I would estimate that a good 10mph of CHS is being produced by an optimum take away that makes the "load-lift-drop-squat" sequence happen. The best guys are doing it and they are 140+chs. Most of us are doing it sub-optimally and produce mid 130s CHS. Those that don't do it well, 130 CHS.
So, how does one do this, and what are the drills. Basically it is a break down. One piece take away while feeling enough lateral speed to get your hands away from you and up. If you draw a line up your lead leg diagonally continuing that line straight to the hands, you should feel "lift" at that point. Do that move until you are comfortable and do it until you feel the club working WITH you for more lift. If you pre-hinge the club, you probably won't feel it.
Once you have that move down, add the next motions of letting the hands go above/behind your head and let the club set while you feel the "drop" of your body. Timing the drop of your body and the drop of the club head into max shaft angle is critical. I think the club shaft sets and the momentum of the club head feels like it assists the drop speed of the body and pushes your lead foot down. It should then feel like "go time" once you hit the bottom and time for a massive squat-push leg drive that now has inertial "traction" for the big "jump' right before impact.
If you have trouble "feeling" these forces from the velocities, use a baseball bat. It is a twice as heavy, no flex, but the MOI of the bat is closer to the hands and does not slow the sequence down much. I feel the best is little league baseball bat. 24 oz at 28 inch long. You should feel it then.