Any plyometric exercises I can do without equipment? Want to improve kicking and punching speed

For punches:
Do the "Howitzer Gamble":
Push ups on fists, feet elevated by at least 12 inches. Down to at least 4" off ground every rep but deeper is better.
"As Many Reps As Possible", 30s rest, AMRAP, 30s rest, AMRAP, 30s rest, AMRAP, 2minute rest, AMRAP, 2min rest, AMRAP, 2min rest, AMRAP, 10min rest, AMRAP, 10min rest, AMRAP, 10min rest, AMRAP, 30min rest, AMRAP, 30min rest, AMRAP, 30min rest, AMRAP, 10min rest, AMRAP, 10min rest, AMRAP, 10min rest, AMRAP, 2min rest, AMRAP, 2min rest, AMRAP, 2min rest, AMRAP, 30s rest, AMRAP, 30s rest, AMRAP.
I'm sure you see the general pattern.

You can do 3 more sets of AMRAP with 30 second rests if you want. It is assumed you will limit yourself out of mental fatigue but push yourself every set and each day best you can. This will build mind-body connection because being told to go to failure is all mental fatigue and strength, and the increasing then decreasing rest times really helps in building mind-body. Do this every day.

After a month start doing 1, 000 punches every day in one session. Every week add 500 punches. After you reach 10, 000 punches just keep at it every day. Go ahead and incorporate footwork. Do your best.

For legs I suggest kneesovertoesguy 7 mobility checklist exercises, nordic curl, split squat, and foot lift. You can find the videos in the Body Toughening thread I made. Set a baseline and be consistent in general volume and weight used but adjust reps and sets as you see fit. Just watch his videos man they're great.
After a month your leg muscles will develop, your connective tissue will have developed, and your mind-body connection a bit. You want to keep progressing this but start doing your kicks for high repetitions about 1, 000 total as well. Progress each week by 500 until you reach 10, 000 and be consistent from there.

That's about all you need to do. Speed is repetition and mind-body connection from endurance training.
 
For punches:
Do the "Howitzer Gamble":
Push ups on fists, feet elevated by at least 12 inches. Down to at least 4" off ground every rep but deeper is better.
"As Many Reps As Possible", 30s rest, AMRAP, 30s rest, AMRAP, 30s rest, AMRAP, 2minute rest, AMRAP, 2min rest, AMRAP, 2min rest, AMRAP, 10min rest, AMRAP, 10min rest, AMRAP, 10min rest, AMRAP, 30min rest, AMRAP, 30min rest, AMRAP, 30min rest, AMRAP, 10min rest, AMRAP, 10min rest, AMRAP, 10min rest, AMRAP, 2min rest, AMRAP, 2min rest, AMRAP, 2min rest, AMRAP, 30s rest, AMRAP, 30s rest, AMRAP.
I'm sure you see the general pattern.

You can do 3 more sets of AMRAP with 30 second rests if you want. It is assumed you will limit yourself out of mental fatigue but push yourself every set and each day best you can. This will build mind-body connection because being told to go to failure is all mental fatigue and strength, and the increasing then decreasing rest times really helps in building mind-body. Do this every day.

After a month start doing 1, 000 punches every day in one session. Every week add 500 punches. After you reach 10, 000 punches just keep at it every day. Go ahead and incorporate footwork. Do your best.

For legs I suggest kneesovertoesguy 7 mobility checklist exercises, nordic curl, split squat, and foot lift. You can find the videos in the Body Toughening thread I made. Set a baseline and be consistent in general volume and weight used but adjust reps and sets as you see fit. Just watch his videos man they're great.
After a month your leg muscles will develop, your connective tissue will have developed, and your mind-body connection a bit. You want to keep progressing this but start doing your kicks for high repetitions about 1, 000 total as well. Progress each week by 500 until you reach 10, 000 and be consistent from there.

That's about all you need to do. Speed is repetition and mind-body connection from endurance training.

****** !! Before anyone tries this, bear in mind this is from someone who doesn't train martial arts and has just chosen stuff he's seen on the internet.

Please take advice from the actual martial artists who post on here.
 
Want to improve punching and kicking speed
You may try this:

Stay in a horse stance.

1. Raise your right hand next your right ear, chop 45 degree down to the left side of your left knee (loose your shoulder joint).
2. Move your right hand to hit in front of your chest (loose your elbow joint).
3. Move your right hand to the right side of your right ear 45 degree up (loose your wrist joint).
4. Drop your right hand with fingers down next to your right knee (let energy to go through your shoulder joint, elbow joint, wrist joint, and reach to your finger tips).
5. Repeat 1 - 4 for your left hand.

You keep moving like this until you can only hear the wind and your hands are just a blur around your body, you have developed some speed.

This training (break through 3 joints) is from the CMA Zimen system. As far as I know this system can produce very fast hand from this training method.

I can only find this clip. It's 60% similar but not 100% the same.

- The front strike (loose elbow joint) is missing.
- the stretching is not to the maximum.
- The speed is not to the maximum.

 
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Air punches and kicks. I always find punches and kicks in air increase speed and heavy bag work increase power.
 
Want to improve punching and kicking speed
This can get into a very in-depth discussion as there are various elements and components involved and there is no single answer. I'll just list some basic ideas and see where it goes. Do note that these same ideas apply to blocking as well.

1. There is speed in regards to how fast the weapon moves thru space (neuro-muscular), and there is speed in terms of position - where your fist is in relation to the target. They are two different things. The bottom line is, it's not how fast you go, but how soon you get there that counts. (I've posted on this in prior threads)

2. One common mistake is loading up, cocking the hand back just before the punch. As you draw back you are going in the wrong direction, as well as telegraphing. This is a natural habit (that is very difficult to self-diagnose, hence the need for an instructor's continual feedback) but can be broken with practice.

3. Power is good in a punch, but sometimes the method of its generation is at the expense of speed. Using the sequential engagement of muscles (power starting from the legs > hips > shoulder > arm) using a whipping motion is strong, but takes time to build up to maximum power. Against a slower reacting opponent, this is OK., but the extra power is at the loss of some speed.

4. Moves must be spontaneous for max speed. Power generation has to be more sudden than described above in #3 - the whole body has to explode at one time. This is a more advanced concept, I think, and hard to explain in a few words. This IMO is a good balance of speed and power.

5. One speed technique (that Ed Parker explained to me) is instead of the feeling of pushing the punch out, think of it being pulled out, or think of your arm as an arrow in a drawn bow that jumps out as you loose it. I feel this does increase speed, but at some loss of power.

6. Re: kicking speed - legs are heavier than arms and leg muscle power is needed. I see many kicks being extensions of knee kicks. Get that knee up quicker, relax the joint, and the foot will snap out quicker. So I would work on knee speed drills and on leg and hip strength. Some of the other numbered ideas can be applied as well.

7. My style mostly uses fast snapping kicks. I was taught, "Kick the foot out at 100 mph, snap it back at 200 mph." Somehow this mental concept makes my kicks faster.

These are just some thoughts off the top of my head. Others have suggested various specific exercises, and there has been some good discussion in the past on fast twitch muscle physiology and development, but I wanted to put forth some ways of looking at the general topic of speed from different perspectives.
 
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This is in the "for-what-it's-worth" column but here it goes. There is a substantial "mental" element to speed. First you must practice being relaxed. I tell my students to think of their arms/legs as whips rather than rigid battering rams. Along with that I urge them to concentrate on bringing the punch/kick back instead of on getting it to the target. When punching I envision the fist already on the target even before the punch is thrown. An analogy would be the (admittedly a tired, cliched example) snatching the coin from the teacher's hand. Without total relaxation the arm will move slowly and sacrifice power.
 
It is hard to make suggestion as OP have not told people what is his level. If OP is a beginner, concentrate on how to punch and kick first, learn to use the foot, hip, waist and shoulder to generate power and the limb relax like a whip first. Don't worry about speed.

If OP has gone through all the basics, then it's a different story. Speed is not the only thing, learn NOT to telecast your movement is just as if not more important. Reaction time of opponent is from the time he/she notice the attack to the time of hit, you can be very fast in striking, but if you telecast 1/2 sec before striking, all the speed doesn't mean a thing.

It was funny long time ago, we play with one person just hold the palm out, the other person try to hit the palm by a punch or anyway just to touch the palm. The person that held the palm out try to move the palm away from being touch. One time, there was a girl that never trained before, she manage to hit everyone over and over. She's slow, nothing fast and she was not even trying to "whip" our hands. She just didn't have any telecast. It was the hardest to get away!!! We all red faced!!!

Telecast can be anything. I played with my grand daughter on striking just last week. She said she can tell by looking at my face, my expression change before striking. She said my facial change before I strike!!!
 
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