Variable velocity in two dimensions Problem set B Answers

1. (a) \(\mathbf{a} = \begin{pmatrix}8t - 7\\-9t^{2}\end{pmatrix}\); \(\mathbf{a}(3) = \begin{pmatrix}17\\-81\end{pmatrix}\) m s⁻²
2. (b) \(\mathbf{a} = \begin{pmatrix}20e^{4t}\\-27e^{-3t}\end{pmatrix}\); \(\mathbf{a}(0.4) = \begin{pmatrix}99.1\\-8.13\end{pmatrix}\) m s⁻²
3. (c) \(\mathbf{a} = \begin{pmatrix}-15\sin 3t\\27\cos 3t\end{pmatrix}\); \(\mathbf{a}(1.5) = \begin{pmatrix}14.7\\-5.69\end{pmatrix}\) m s⁻²

1. (a) \(\mathbf{r} = \begin{pmatrix}-\frac{t^{3}}{3} + 4t^{2} - 2\\t^{3} + 4t + 5\end{pmatrix}\)
2. (b) \(\mathbf{r} = \begin{pmatrix}4e^{2t} - 3\\7 - 3e^{-3t}\end{pmatrix}\)
3. (c) \(\mathbf{r} = \begin{pmatrix}5 - 2\cos 5t\\3\sin 2t - 1\end{pmatrix}\)

1. (a) 23.5 m s⁻¹
2. (b) 30.4 m s⁻¹
3. (c) 11.1 m s⁻¹

7.35 m s⁻¹

1. (a) 48.4°
2. (b) \(\mathbf{a} = 2\mathbf{i} - 6\mathbf{j}\), 6.32 m s⁻²

1. (a) 16.1 m
2. (b) 18.1 m

1. (a) \(\mathbf{a} = \begin{pmatrix}3\\-0.797\end{pmatrix}\) m s⁻²
2. (b) \(\mathbf{r} = \begin{pmatrix}-3.63\\6.80\end{pmatrix}\) m
3. (c) 7.71 m
4. (d) 0.639 m s⁻¹

1. (a) 15 m s⁻¹
2. (b) \(\mathbf{r} = \begin{pmatrix}-4.52\\2.14\end{pmatrix}\) m
3. (c) 5 m. The speed and the distance from the origin are the same at every time, so the path is a circle about the origin.

least speed 1.34 m s⁻¹ at \(t = 0.896\)

\(t = 0.524\) s; speed 5 m s⁻¹