Variable velocity in two dimensions Problem set A Answers

1. (a) \(\mathbf{a} = \begin{pmatrix}10t - 4\\-6t^{2}\end{pmatrix}\); \(\mathbf{a}(2) = \begin{pmatrix}16\\-24\end{pmatrix}\) m s⁻²
2. (b) \(\mathbf{a} = \begin{pmatrix}21e^{3t}\\-24e^{-2t}\end{pmatrix}\); \(\mathbf{a}(0.5) = \begin{pmatrix}94.1\\-8.83\end{pmatrix}\) m s⁻²
3. (c) \(\mathbf{a} = \begin{pmatrix}32\cos 4t\\-24\sin 4t\end{pmatrix}\); \(\mathbf{a}(1.2) = \begin{pmatrix}2.80\\23.9\end{pmatrix}\) m s⁻²

1. (a) \(\mathbf{r} = \begin{pmatrix}-\frac{t^{3}}{3} + 3t^{2} + 1\\t^{4} + 2t - 3\end{pmatrix}\)
2. (b) \(\mathbf{r} = \begin{pmatrix}3e^{3t} - 1\\9 - 4e^{-2t}\end{pmatrix}\)
3. (c) \(\mathbf{r} = \begin{pmatrix}2\sin 3t - 1\\7 - 4\cos 2t\end{pmatrix}\)

1. (a) 20.6 m s⁻¹
2. (b) 24.7 m s⁻¹
3. (c) 9.80 m s⁻¹

9.93 m s⁻¹

1. (a) 66.8°
2. (b) \(\mathbf{a} = 3\mathbf{i} - 5\mathbf{j}\), 5.83 m s⁻²

1. (a) 23.4 m
2. (b) 5.66 m

1. (a) \(\mathbf{a} = \begin{pmatrix}2\\-0.812\end{pmatrix}\) m s⁻²
2. (b) \(\mathbf{r} = \begin{pmatrix}-5\\3.19\end{pmatrix}\) m
3. (c) 5.93 m
4. (d) 1.29 m s⁻¹

1. (a) 12 m s⁻¹
2. (b) \(\mathbf{r} = \begin{pmatrix}5.91\\1.02\end{pmatrix}\) m
3. (c) 6 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 4 m s⁻¹ at \(t = 2.00\)

\(t = 0.785\) s; speed 8 m s⁻¹