There have been a few situations, though thankfully few and far between, where a question about how to deal with some of the more extreme aspects of physics comes into play. By this I mean there have been a number of situations in which characters have exceeded the bounds of normal everyday physics that we are used to dealing with. In particular issues of extreme speed, extreme hot or cold, and the like.
Speed
Everyone like to have characters able to move quickly. In particular, fans of many Anime works where it shows the characters going absurdly fast. We have nothing against characters going extremely fast or even absurdly fast, they've got plaid!
Sound / Sonic Booms
There are some consequences for going that fast, however. The prime factors to consider are the sound barrier (achieving Mach speeds) as well as acceleration and deceleration. This is kind of important, especially with Mach speeds (Mach 1, for those that do not know, is the speed of sound). Achieving Mach 1 creates a sonic boom[1]. Sonic booms are very, very loud shockwaves. They have been known to shatter windows on the ground from an aircraft flying overhead. This means that if a character goes Mach 1 or above they are going to create this loud pressure shock wave.
A sonic boom indoors is catastrophic to anyone with a normal biological body. Consider what happens in an enclosed space if someone drops something very heavy on a hard concrete floor. The sound echoes off the walls and is amplified, thus not only is it louder to those nearby but the sound travels farther. The same is true for a sonic boom made indoors, the pressure wave is going to damage the hearing ability of anyone nearby. Glass surfaces will likely shatter, it may even damage the soft tissue of the brain or eyes in the process.
Acceleration / Deceleration
There is a handy and well known physics equation about force.[2]
force=mass*acceleration
This is pretty handy to know, because it means that a character has to put the energy into speeding up, duh. No one hops in their car and it is instantly at freeway speeds. Airplanes have special shock absorption in seats for when the airplane takes off and accelerates to cruising velocity and altitude. So if a character can go, say, 100 meters per second (over 220 miles per hour), they have to reach that speed first. In terms of Star Trek, they have inertial dampeners on ships that mean one can go to warp and not kill everyone on the ship from g-forces. Accelerating to 100 meters per second requires enough room to do so. Keep in mind, the land speed record isn't as much an issue of making a faster engine, but finding a long stretch of land that one can build up the speed. Advancements in engine design make building up speed easier (faster) and thus one can get a higher speed, but it takes a lot of room.
So where does force come into things? Well, say a character is accelerating to the above mentioned Mach 1. The math gets complicated very quickly (uses calculus and integration to calculate), but suffice it to say the quicker someone wants to actually get to Mach 1 the more force is required and thus the more force their body will carry with it. The question then quickly becomes, what if that person wants to turn? What if they trip, stumble, or anything else of that kind?
The way speed (velocity) works with inertia means that if someone is going 100 meters/second going in a direction, we'll say going North. Physics says that velocity has a direction to it, a direction that some object with mass (someone's body) is moving in. Inertia says that said body is going to keep going in that direction until acted upon by some outside force. Watch hockey some time, when someone slams into the wall, that is a perfect illustration of velocity and inertia. Same with NASCAR, when a car on the track is going fast enough and they try to turn too sharply, the inertia forces the car to continue moving in the direction it started in. Since the wheels are no longer pointing in that direction, the force has to go somewhere and it rolls the car.
So now, getting back to the person whose trying to accelerate to Mach 1. The short answer is if the person tries to turn, they're going to have to overcome that inertia, counteract the velocity pointing in one direction and redirect that into another direction. This is what makes a person lean to one side when turning their car around a tight corner. The friction of the wheels along with the weight of the car help stop the car from going on in one direction, while the wheels turning in the new direction give a place for that force to go. Tripping is equally problematic, considering it means that one would hit the ground with a force equal to how quickly they go from moving at Mach 1 (or near that) to 0. This is what causes a runner to roll if they trip and fall over on the track.