Why is the frictional force on a wagon’s wheel in the opposite direction from the frictional force on a car’s wheel?

When you pull a wagon forward, friction from the ground starts the wheel turning and it does this by pushing backward on the bottom of the wheel. Friction is thus preventing the wheel from skidding across the pavement. When you step on a car’s accelerator, the car’s engine starts the wheel turning and friction from the ground pushes forward on the bottom of the wheel to prevent the wheel from skidding across the pavement. In the first case, friction is trying to help the wheel to turn while in the second case friction is trying to keep the wheel from turning. That’s why the forces (and the resulting torques) on the wheel are in opposite directions for the two cases.

Why is it that we can use energy without doing work? Where does this energy go? For example, you could push on a wall until your arms fell off, but you wouldn’t have done any work.

When you are pushing on something without doing any work, your energy is being converted directly into thermal energy inside your body. Your muscles are inefficient and they convert food energy into thermal energy whenever they are under tension. It’s like a car, which uses gasoline even when it’s stopped at the light. The engine keeps running but it does no work. Similarly, if you simply burned your cereal in your breakfast bowl, you would turn its energy directly into thermal energy without doing any useful work. Your body is also able to burn up that food energy and create thermal energy, albeit a little less visibly.

Why are tires filled with air instead of something less likely to go flat?

This is an interesting question with several answers. First, a solid rubber tire would have a huge mass and would require consider work to accelerate. Because it rotates as the car moves, a tire stores twice as much kinetic energy as the other parts of the cars. By reducing the mass of the tires, the car reduces the amount of energy it must put into the tires to get them moving and the amount of energy it must remove from the tires to stop them from turning.

Secondly, a solid rubber tire would be so hard that it would give the car a very rough ride. The air in the tires cushions the car against many of the rough spots it drives over. Without the air cushion, the wheels and axles would bound up and down with every pebble in the road.

Lastly, a solid rubber tire would be very expensive. The materials used in a tire are expensive and a tire’s cost should be roughly proportional to its weight. Since a solid tire would weigh much more than an air-filled one, it would also cost much more. Its tread would still wear out, so it wouldn’t last any longer than an air-filled tire.