A hardware store has two routes up to its 2.0 m storage shelf: a straight ladder, and a long ramp that wraps around the aisle. Vada carries a 50 N box up the ladder. Wells wheels an identical 50 N box up the ramp.
The ramp route covers three times the distance. Which box ends up with more stored energy?
Track the heights. Both boxes start on the floor and finish on the same 2.0 m shelf.
The height gained is 2.0 m for each box, whichever route it took.
Stored energy is weight × height gained. For each box: 50 × 2.0 = 100 J.
The ramp's extra distance adds travel, not height. The GPE change reads the height, nothing else.
A winding staircase, an elevator, a straight ladder: up to the same floor, the same GPE change.
The reverse holds too. A ball dropped straight down and a ball rolled down a long ramp to the same floor release the same GPE.
Xenia's cat reaches the same 1.2 m windowsill two ways: a single straight jump, or a long walk up a carpeted pet ramp. How do the cat's two GPE gains compare?
A mover hauls a 300 N dresser straight up on a rope to a second-floor window, 4.0 m up. A second identical dresser goes up the stairwell, covering 12 m of stairs to the same window. Compare the two dressers' GPE gains.
Both dressers gain the same 4.0 m of height.
The GPE change reads the height change only.
Each gain: 300 × 4.0 = 1200 J.
Both dressers gain 1200 J; the stair route changes the travel, not the height.
Two hikers climb from a trailhead to a lookout 150 m higher. One takes a steep 400 m trail; the other takes a gentle 900 m trail. Each carries an 80 N pack. Compare the packs' GPE gains.
Each pack gains the same 150 m of height.
Each gain: 80 × 150 = 12000 J.
Both packs gain 12000 J; the longer trail does not change the height.
Why the path cannot matter is the next lesson's question.
A 1.5 kg package rides a winding conveyor from the floor up to a 3.0 m rack. The conveyor's winding belt covers 9.0 m. Take g = 9.8 N/kg. Enter the package's GPE gain, in J.
Two window cleaners haul identical 60 N buckets to the same 5.0 m balcony: one straight up on a rope, the other zigzagging up scaffolding and covering 14 m. Which quantity sets each bucket's GPE gain?
Four identical 40 N toolboxes are moved from a shop floor to the same 2.5 m mezzanine: one by elevator, one by ladder, one by a long ramp, and one by a winding stair. Which toolbox gains the most GPE?
A roofer's 20 kg shingle bundle rides a lift cart up a sloped 8.0 m track to a roof edge 6.0 m above the ground. Take g = 9.8 N/kg. Enter the bundle's GPE gain, in J.
Three boxes are raised from the floor by different routes: a 100 N toolbox through 1.0 m, a 100 N paint crate through 2.0 m, and a 100 N tile box through 3.0 m. Which box gains the most GPE?