Halliday, Resnick & Walker — Motion in Two and Three Dimensions: Problem 114

The position vector \(\vec{r}\) of a particle moving in the \(xy\) plane is \(\vec{r} = 2t\hat{i} + 2 \sin[(\pi/4 \text{ rad/s})t]\hat{j}\), with \(\vec{r}\) in meters and \(t\) in seconds. (a) Calculate the \(x\) and \(y\) components of the particle’s position at \(t = 0, 1.0, 2.0, 3.0,\) and \(4.0 \text{ s}\) and sketch the particle’s path in the \(xy\) plane for the interval \(0 \leq t \leq 4.0 \text{ s}\). (b) Calculate the components of the particle’s velocity at \(t = 1.0, 2.0,\) and \(3.0 \text{ s}\). Show that the velocity is tangent to the path of the particle and in the direction the particle is moving at each time by drawing the velocity vectors on the plot of the particle’s path in part (a). (c) Calculate the components of the particle’s acceleration at \(t = 1.0, 2.0,\) and \(3.0 \text{ s}\).