Worked Exercise — Imola-Lugo Route

Problem

From Imola to Lugo you travel a stretch of $6{,}29\;\text{km}$ in a (roughly) north-east direction, then a stretch of $4{,}8\;\text{km}$ due east. Choosing an axis system with $x$ pointing east and $y$ pointing north, find the overall displacement vector.

Solution

Step 1 — Resolve the first stretch. Assuming the first stretch is at $45^\circ$ to east, its components are

$\vec{a} = (6{,}29\cos 45^\circ,\ 6{,}29\sin 45^\circ) \approx (4{,}45;\ 4{,}45)\;\text{km}$

Step 2 — The second stretch. Directed purely east, it has only an $x$ component:

$\vec{b} = (4{,}8;\ 0)\;\text{km}$

Step 3 — Sum component by component. The overall displacement is

$\ev{\vec{a}+\vec{b} = (9{,}25;\ 4{,}45)\;\text{km}}$

Step 4 — Magnitude of the displacement. The straight-line Imola-Lugo distance is

$\norm{\vec{a}+\vec{b}} = \sqrt{9{,}25^2 + 4{,}45^2} \approx 10{,}3\;\text{km}$

Links

Topics: Vettori Concepts: Componenti di un vettore Skills: Uso della trigonometria · Scomposizione vettoriale Methods: Scomposizione in componenti cartesiane