When you carry an object across the room, without lifting it or setting
it down, you do no "mechanical work" on it.
You need to carry a suitcase up a flight of stairs. In which case will you do
the most mechanical work?
You carry the suitcase up quickly.
You carry the suitcase up slowly.
Both cases involve the same amount of work.
The amount of potential energy possessed by an elevated object is
the distance it is lifted,
the force needed to lift it,
the work done in lifting it, or
the value of the acceleration of gravity.
Ralph asked me a question about today's reading assignment. According
to the reading assignment, a car coasting from rest
down two hills, one steeper than
the other, would arrive at the bottom of each hill with the same speed, as
long as the two hills have the same vertical height. (Of course, this is
true only if we neglect friction and air resistance.) Ralph wondered how
this could be possible, since the acceleration of the car down the steep
hill is much greater than down the other hill. What should you tell him?
The car on the steeper hill will have more acceleration, that is true. However, it will not be accelerating for as long a distance. Viewed from an energy perspective, both start from the same potential energy, so they must both end up with the same kinetic energy. Therefore the two things (steeper, but not as long) exactly cancel out, giving you the same final velocity. (The steeper one will indeed get to the bottom first, though.)