Q.A porter raises a suitcase of mass $15\ \text{kg}$ vertically through $1.5\ \text{m}$ onto a trolley, and then wheels the trolley $10\ \text{m}$ along a level, horizontal platform at constant velocity. Find the work done against gravity in each of the two stages, and explain why the two answers differ so greatly (take $g = 9.8\ \text{m/s}^2$).

Given: mass $m=15\ \text{kg}$, height raised $h=1.5\ \text{m}$, horizontal distance $10\ \text{m}$, $g=9.8\ \text{m/s}^2$.

Raising the suitcase. Here the displacement (vertically upward, $1.5\ \text{m}$) is exactly opposite to gravity's own direction (downward), so the porter, in raising the case, does positive work against gravity equal to the resulting increase in gravitational PE: $W = mgh = 15 \times 9.8 \times 1.5 = 220.5\ \text{J}$

Wheeling the trolley horizontally. Here the suitcase's height above the ground does not change at all -- the entire $10\ \text{m}$ displacement is horizontal, perpendicular to gravity's (vertical) direction. Since work done by (or against) a force depends on the component of displacement along that force's own direction, and the vertical component of a purely horizontal displacement is exactly zero, the work done against gravity during this stage is $W = mgh_{\text{change}} = mg(0) = 0\ \text{J}$

The two answers differ so greatly because work against gravity depends only on the change in height, not on the distance actually travelled: $1.5\ \text{m}$ of vertical rise requires real work against gravity, however short the distance, while $10\ \text{m}$ of purely horizontal travel, however long, requires none at all (though of course some other force -- the porter's push against friction and inertia -- is still needed to keep the trolley moving; that is a separate matter from work done specifically against gravity).

Compute $W=mgh$ for the vertical-lift stage using the actual height risen; recognise the horizontal stage involves zero change in height, so the work done against gravity there is zero.

• Assuming any motion against some resistance counts as "work against gravity", and wrongly computing $mg\times10\ \text{m}$ for the horizontal stage.
• Forgetting that work against gravity depends on height change alone, not on the total path length travelled.