Why on Pg. 6, 2nd full paragraph, it says the car is accelerating if the slope of the road changes but in the “not accelerating” list it says a bicycle going up a hill is not accelerating. Aren’t those the same situation?

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.

Why is force = mass * acceleration an exact relationship (i.e. why not force = 2 * mass * acceleration)?

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.

Why is 45° above horizontal the ideal angle to throw something the greatest distance if gravity is acting on the vertical direction but not the horizontal?

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.

Why does an object accelerate when it changes direction?

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 miles-per-hour, if that feels more comfortable), it heads north at 10 m/s (or 10 miles-per-hour). 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.

Why does a ball fall 4.9 meters during its first second of falling?

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.

Why do you feel no acceleration in free fall, even though you are accelerating?

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.

Why do two objects of unequal mass fall and hit the ground at the same time?

If one object has twice the mass of the other, then it is twice as hard to accelerate. To make it keep pace with the other ball, it must experience twice the force. Fortunately, gravity pulls on it twice as hard (it has twice the weight of the other ball), so in falling, it does keep pace with the other ball. The two fall together. Just for fun, imagine stepping off the high diving board with two friends. The three of you have essentially identical masses and weights and also fall at the same rate. Now imagine that two of you hold hands as you fall. You are now a single object with twice the mass of your other friend. Nonetheless, you still fall at the same rate. So an object with twice the mass of another falls at the same rate as that other object.

Why do objects on earth accelerate downward at the same speed regardless of their mass?

What you mean here is that they accelerate downward at the same rate (“speed” has a particular meaning that isn’t so well suited to discussions of acceleration). This fact comes about because, although massive objects are harder to accelerate, they also experience more weight. Thus a huge stone will fall at the same rate as a small rock because the stone will be pulled downward more strongly by gravity and that extra pull will make up for the stone’s greater inertia.

While gravity supposedly makes all objects accelerate at the same rate, feathers do not seem to comply. What factors affect the feather’s acceleration, besides air resistance (which should affect all objects equally)?

Actually, air resistance doesn’t affect all objects equally. The feather has so much surface area that it pushes strongly on the air through which it moves and the air pushes back. For an object with very little mass and weight, the feather experiences an enormous amount of air resistance and has great difficulty moving through the air. That’s why it falls so slowly. If you were to pack a feather into a tiny pellet, it would then fall just about as fast as other objects. Similarly, you fall much more slowly when your parachute is opened because it then interacts with the air much more effectively.