Michaela is planning a trip in Ireland from Killarney to Cork to visit Blarney Castle. (See Example \(3.2 .\) ) She also wants to visit Mallow, which is located \(39 \mathrm{km}\) due east of Killarney and \(22 \mathrm{km}\) due north of Cork. Draw the displacement vectors for the trip when she travels from Killarney to Mallow to Cork. (a) What is the magnitude of her displacement once she reaches Cork? (b) How much additional distance does Michaela travel in going to Cork by way of Mallow instead of going directly from Killarney to Cork?

We first need to sketch the displacement vectors of Michaela's trip. According to the question, Mallow is \(39km\) to the east of Killarney and \(22km\) to the north of Cork. Let's use standard Cartesian coordinates with the \(x\)-axis as East direction and the \(y\)-axis as North direction. Now, let's represent the displacement vectors: - Killarney to Mallow: \(\vec{A}\), where the \(x\)-component is \(39km\) and the \(y\)-component is \(0km\) (as it is only in east direction). Therefore, \(\vec{A} = 39\hat{i}\). - Mallow to Cork: \(\vec{B}\), where the \(x\)-component is \(0km\) and the \(y\)-component is \(-22km\) (as it is in the south direction). Therefore, \(\vec{B} = -22\hat{j}\).

To find the magnitude of her displacement once she reaches Cork, we need to add the two displacement vectors: \(\vec{A}\) and \(\vec{B}\). The total displacement vector: \(\vec{C} = \vec{A} + \vec{B} = 39\hat{i} - 22\hat{j}\). Now, let's calculate the magnitude of the total displacement vector \(\vec{C}\): \(| \vec{C} | = \sqrt{(39)^2 + (-22)^2} = \sqrt{1765} \approx 42.01km\).

To calculate the additional distance Michaela travels by going to Cork through Mallow instead of going directly from Killarney to Cork, we have to compare the trip from Killarney directly to Cork (magnitude of \(\vec{C}\)) and the trip from Killarney to Mallow and then to Cork (magnitude of \(\vec{A}\) and \(\vec{B}\), respectively). The distance traveled from Killarney to Mallow to Cork is the sum of the magnitudes of \(\vec{A}\) and \(\vec{B}\): \(| \vec{A} | + | \vec{B} | = 39km + 22km = 61km\). The additional distance Michaela travels by going to Cork through Mallow is the difference between the distance along her actual trip and the magnitude of her total displacement: \(Additional \ Distance = | \vec{A} | + | \vec{B} | - | \vec{C} | = 61km - 42.01km \approx 18.99km\). So, Michaela travels an additional distance of approximately \(18.99km\) by going to Cork through Mallow instead of going directly from Killarney to Cork.