I keep seeing folks having questions about how to warm up... and sadly, a lot of misinformation on what it does or how you should do it.
How Muscle Tears Happen
Your body has certain safeties built into it. One of them is that your body is designed to let your muscles take damage before your joints take damage- muscles heal easier than joints.
In each of your muscle bundles, there are "Golgi Tendon Reflex" nerves. These things measure how fast your muscles are moving relative to a joint. They exist to activate your muscles to force a "braking action" if something is moving fast enough it could damage the joint.
For example, kicking so hard your knee could flip backward.
Instead of having that, your body activates the antagonistic muscles (the muscles opposite of the movement) to lock up, as quick as possible. This can result in strains or full tears. This response happens at the spinal column- the nerve impulse doesn't even get up to the brain, so this isn't a conscious choice you can make. It's like the response to pull your hand out of a fire before you even feel the pain, same concept.
A warm up is introducing your body to the motions, range of motion, and speed you expect it to perform at. You are resetting the Golgi reflexes to accept a wider range, higher speed than they'd normally be at.
In other words, you're turning off some safeties.
It means your muscles won't do that pull-tear thing, but it also means your muscles are expecting your brain to pay more attention and conscious control around not letting the joint move outside of acceptable ranges.
In order for a warm up to help your performance, it must involve similar or identical motion to what you intend to do.
You start light in terms of speed and force and increase to full speed and force, thereby getting the muscle spindles to turn down the instinctual responses and accept new settings. Using similar movement also means that blood is going to the same muscles that you will be using.
You may need to do some minimal stretching on your antagonistic muscles if they generally tend to be tight from stuff like, you know, sitting in an office all day, etc. - just enough that they're not impeding the primary movement.
What doesn't help
Doing unrelated movements
If you're going to do upper body weight training, jogging doesn't really help. Sure, it's getting you some cardio, and maybe loosing up some shoulder stuff from the bouncing and arm swings... but yeah, not specifically helping you.
Too much range of motion (NEVER DO THIS)
If you warm up by giving yourself a lot of stretches or movements that take your range of motion -greater- than what you plan on doing... you're setting yourself up for injury.
Remember how I said your resetting your muscle fibers to accept a bigger motion and turn off the safeties? When you reset it to take a much bigger range than what you're doing, you're turning off your stabilizing muscles - and increasing the odds of a joint injury.
Why does this/that/the other source say to do jogging/jumping jacks/stretches/etc.?
Getting literally "warmed up" feels good. It gets your brain and nervous system into, "Hey, we're going to DO STUFF" mode.
Thing is, not every activity needs the same kind of warm up. I'm thinking a lot of "warm up advice" is partially a push to get more people to do cardio. The other part of it is assuming people are doing general workouts which a general warm up makes sense.
But it's so simplified and generic, it's equivalent to saying "Take asprin" for any given medical condition - great when you have a minor ache or pain, not so great for many other conditions.
Which is why we keep seeing studies coming out about whether warming up or stretching actually prevents injuries or not. It's highly contextual on the activity and the type of warm up the person did- some warm ups will help certain activities, some will increase your odds of injury.
So, you reset your muscle spindles to do stuff, right? Might be a good idea to reset them again to go back to doing normal things.
If you were doing power/endurance training, it might be a good idea to gently stretch things to keep them from tightening up and cutting off blood flow. If you were doing lots of cardio, maybe slow down the body to normal walking and get it used to that idea, stretch a few things and swing around whatever limbs were being used less- get the blood a bit more evened out.