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Your example of pushing the car leads to my explanation to younger students (grade school) differentiating mass and weight. They all remember pulling a wagon, so I ask them if it is easy to pull an empty wagon on a level sidewalk and they all agree that it would be. Then I ask them to imagine 5 of their buddies hopping into the wagon and they giggle at the image. I ask them to imagine pulling the wagon with that many children in it and whether it would be harder to pull than when it was empty. They can all agree it would be more difficult. I tell them that it is more difficult to move because of the additional MASS of the added bodies, not the weight as they are not trying to lift the wagon. Of course actually doing the activity would be better, but I usually have just an hour with them to give them a taste of "all" of physics.
Pati