

This wonderful question has many answers. The first, and most direct, is that you do feel the acceleration. You feel an upward fictitious force (not a real force at all, but an effect of inertia) that exactly balances your downward weight. The feeling you experiences is "weightlessness." That's why your stomach feels so funny. You're used to having it pulled downward by gravity but the effect of your fall is to make it feel weightless.
As a simple argument for that result, think about the ball's speed as it falls: it starts from rest and, over the course of 1 second, it acquires a downward speed of 9.8 m/s. Its average speed during that first second is half of 9.8 m/s or 4.9 m/s. And that is just how far the ball falls in that first second: 4.9 m. By holding the ball 4.9 m above the floor, you can arranged for it to hit one second after you drop it.
What you mean by "changes direction" is that the direction part of its velocity changes. For example, instead of heading east at 10 m/s (or 10 milesperhour, if that feels more comfortable), it heads north at 10 m/s (or 10 milesperhour). This change in direction involves acceleration. The car must accelerate toward the west in order to stop heading east, and it must accelerate toward the north in order to begin moving north. Actually, it probably does both at once, accelerating toward the northwest and shifting its direction of motion from eastward to northward.
The 45° angle is ideal because it gives the ball a reasonable upward component of velocity and also a reasonable downfield component of velocity. The upward component is important because it determines how long the ball will stay off the ground. The downfield component is important because it determines how quickly the ball will travel downfield. If you use too much of the ball's velocity to send it upward, it will stay off the ground a long time but will travel downfield too slowly to take advantage of that time. If you use too much of the ball's velocity to send it downfield, it will cover the horizontal distances quickly but will stay of the ground for too short a time to travel very far. Thus an equal balance between the two (achieved at 45°) leads to the best distance. Note that this discussion is only true in the absence of air resistance.
The answer to this puzzle lies in the definition of force. How would you measure the amount of a force? Well, you would push on something with a known mass and see how much it accelerates! Thus this relationship (Newton's second law) actually establishes the scale for measuring forces. If your second relationship were chosen as the standard, then all the forces in the universe would simply be redefined up by a factor of two! This redefinition wouldn't harm anything but then Newton's second law would have a clunky numerical constant in it. Naturally, the 2 is omitted in the official law.
Here is why the two situations are different:
In the first case, the car is traveling on a road with a changing slope. Because the road's slope changes, the car's direction of travel must change. Since velocity includes direction of travel, the car's velocity must change. In short, the car must accelerate. Picture a hill that gradually becomes steeper and steeper—the car's velocity changes from almost horizontal to almost vertical as the slope changes. In the second case, the bicycle is climbing a smooth, straight hill at a steady speed. Since the hill is smooth and straight, its slope is not changing. Since the bicycle experiences no change in its direction of travel or its speed, it is traveling at a constant velocity and is not accelerating.
There are two levels at which to work. First, there is Newtonian gravity—an attraction that exists between any two objects and that pulls each object toward the center of mass of the other object with a force that's equal to the gravitational constant times the product of the two masses, divided by the square of the distance separating the two objects. For example, you are attracted toward the earth's center of mass with a force equal to the gravitational constant times the product of the earth's mass and your mass, divided by the square of the distance between the earth's center of mass and your own center of mass. This force is usually called "your weight." The earth is attracted toward your center of mass with exactly the same amount of force.
Second, there is the gravity of Einstein's general relativity—a distortion of space/time that's caused by the local presence of mass/energy. Space is curved around objects in such a way that two freely moving objects tend to accelerate toward one another. As long as those objects aren't too large or too dense, this new description of gravity is equivalent to the Newtonian version—they both predict exactly the same effects. But when one or both of the objects is extremely massive or very dense, general relativity provides a more accurate prediction of what will happen. In reality, mass/energy really does warp space/time and general relativity does provide the correct view of gravity in our universe. The next level of theory, quantum gravity (which will reconcile the theory of general relativity with the theory of quantum physics), is still in the works.
Actually, it was Galileo who first realized that objects have this tendency to continue moving at a steady rate in a straightline path—what we call "inertia." He deduced this fact by studying the motions of balls on ramps. He noted that a ball rolling down a slight incline steadily picked up speed while a ball rolling up a slight incline steadily lost speed. From these observations he realized that a ball rolling along a level surface would roll at a steady speed indefinitely, where it not for friction and air resistance. He was aware that friction, air resistance, and gravity were disturbing the natural motions of objects and had figured out a way to see beyond them. But it wasn't until Newton took up this sort of study that the idea of forces and their effects was properly developed. Overall, it took almost two thousand years, from Aristotle to Newton, for the incorrect idea that objects tend to remain stationary when free of forces to be replaced with the correct idea that objects tend to continue at constant velocity when free of forces.
You're exactly right. Occasionally one of those cartoons shows the coyote falling with the anvil directly above his head and the distance between them remaining constant, which is what should happen (ignoring air resistance). But more often, the coyote falls much faster than the anvil, hits the ground first, and is then pounded by the anvil. It sure would be neat to live in a cartoon—the laws of physics just wouldn't apply.
Decelerating is a very specific acceleration—always in the direction opposite your velocity. If you were heading north and accelerated toward the east, your velocity would soon point toward the northeast. It would have some northward aspect because you were initially heading north and hadn't yet accelerated toward the south. It would have some eastward aspect because you had initially been heading neither eastward nor westward and had since accelerated toward the east.
On the other hand, if you were heading north and then turned toward the east, you would have lost your northward velocity and obtained an eastward velocity. This "turning" would have involved a southward acceleration (to get rid of the northward velocity) and an eastward acceleration (to acquire an eastward velocity).
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