Serway & Jewett — Motion in Two Dimensions: Problem 44

A projectile is launched from the point \((x = 0, y = 0)\), with velocity \((12.0\hat{i} + 49.0\hat{j}) \text{ m/s}\), at \(t = 0\). (a) Make a table listing the projectile's distance \(|\vec{r}|\) from the origin at the end of each second thereafter, for \(0 \leq t \leq 10 \text{ s}\). Tabulating the \(x\) and \(y\) coordinates and the components of velocity \(v_x\) and \(v_y\) will also be useful. (b) Notice that the projectile's distance from its starting point increases with time, goes through a maximum, and starts to decrease. Prove that the distance is a maximum when the position vector is perpendicular to the velocity. Suggestion: Argue that if \(\vec{v}\) is not perpendicular to \(\vec{r}\), then \(|\vec{r}|\) must be increasing or decreasing. (c) Determine the magnitude of the maximum displacement. (d) Explain your method for solving part (c).