What do you mean by "a relaxed manner"? Karate teaches to tense at the moment of impact, not to remain relaxed the whole time. You have to be tense when you strike or your strike has no power. There's no such thing as fully relaxed strike, every strike uses some muscles, and muscles only work by contracting.
We're about to get theoretical.
The long term goal is to deliver your techniques in the most relaxed manner possible with minimial muscles tensing except at the moment of contact. It's already been shown that premature tensing lessens power. Additionally, your muscles will naturally contract/tense at the end of a technique to keep your limbs within their natural range of motion. So, the majority of your muscle use should be in the core with much less use at the extremities.
Part of the problem with thinking about muscle use in technique is that muscles can contract in opposing manners during the same technique thus reducing power.
Imagine the human body with no muscles, no tendons, ligaments, etc anywhere except in the core. A skeleton with abs, lol. How would the core have to move to extend the arm or the leg? A twist of core would twist the upper body then the shoulder. The shoulder's momentum would launch the upper arm out. At that point, the upper arm's momentum would extend the forearm and the hand will travel towards a target. At that moment, add back in the muscles and tendons and form a tensed fist. That's the level of relaxed technique that I'm talking about.
I'm sure you can already see how the MT kick uses a lot of the same priniciples and the way the karate kick is taught violates them. If there were no muscles on the skeleton's leg, you couldn't force the lower leg extension.
Instead, the inital core rotation would have to be so violent that the upper leg shoots out drawing the lower leg into a chambered position. The "turning over the hip" movement, that most MT students are familiar with, ensures that the body's momentum continues through a change of direction that, whip-like, launches out the lower leg. Again, add back the muscles and tense enough to get through contact before relaxing as you finish the technique.
Another example: Throw a reverse elbow and let the momentum extend your arm. Now, throw a pure back fist. That's the difference in relaxation and muscle use. It may not feel like it but the back elbow that turns into a back fist is the faster/more powerful technique.
Obviously, it takes years of practice to reach that level of relaxed, but powerful, technique. But it's what I, as a karateka, should be working towards. I don't have enough MT experience to know if it the same goal exists for them.
