Variable velocity in two dimensions Problem set B

Find the acceleration vector at the given time. Velocities are in m s⁻¹, time in seconds.

1. (a) \(\mathbf{v} = \begin{pmatrix}4t^{2} - 7t\\5 - 3t^{3}\end{pmatrix}\), at \(t = 3\)
2. (b) \(\mathbf{v} = \begin{pmatrix}5e^{4t}\\9e^{-3t}\end{pmatrix}\), at \(t = 0.4\)
3. (c) \(\mathbf{v} = \begin{pmatrix}5\cos 3t\\9\sin 3t\end{pmatrix}\), at \(t = 1.5\)

Find an expression for the position vector \(\mathbf{r}\) in metres. Velocities are in m s⁻¹, and \(\mathbf{r}_0\) is the position at \(t = 0\).

1. (a) \(\mathbf{v} = \begin{pmatrix}-t^{2} + 8t\\3t^{2} + 4\end{pmatrix},\ \mathbf{r}_0 = \begin{pmatrix}-2\\5\end{pmatrix}\)
2. (b) \(\mathbf{v} = \begin{pmatrix}8e^{2t}\\9e^{-3t}\end{pmatrix},\ \mathbf{r}_0 = \begin{pmatrix}1\\4\end{pmatrix}\)
3. (c) \(\mathbf{v} = \begin{pmatrix}10\sin 5t\\6\cos 2t\end{pmatrix},\ \mathbf{r}_0 = \begin{pmatrix}3\\-1\end{pmatrix}\)

Find the speed at the given time. Velocities are in m s⁻¹, time in seconds. Your calculator should be in radians.

1. (a) \(\mathbf{v} = \begin{pmatrix}2t^{2} - 9t\\4t + 3\end{pmatrix}\), at \(t = 5\)
2. (b) \(\mathbf{v} = \begin{pmatrix}5e^{3t}\\11e^{-2t}\end{pmatrix}\), at \(t = 0.6\)
3. (c) \(\mathbf{v} = \begin{pmatrix}12\sin 2t\\5\cos 2t\end{pmatrix}\), at \(t = 1\)

A bee flies in two dimensions in a greenhouse. Its position vector is \(\mathbf{r} = \begin{pmatrix}2t^{2} - 5t\\8\cos 3t\end{pmatrix}\) metres at time \(t\) seconds. Find its speed when \(t = 2\).

A hot-air balloon is modelled in a vertical plane. Its velocity at time \(t\) seconds is \(\mathbf{v} = (8 + 2t)\mathbf{i} + (9 - 6t)\mathbf{j}\) m s⁻¹.

1. (a) Find the angle the initial velocity makes with \(\mathbf{i}\).
2. (b) Show that the acceleration is constant and find its magnitude.

Find the distance from the origin at the given time. Displacements are in metres, time in seconds.

1. (a) \(\mathbf{r} = \begin{pmatrix}t^{2} + 5t\\8 - 2t^{3}\end{pmatrix}\), at \(t = 2\)
2. (b) \(\mathbf{r} = \begin{pmatrix}t^{2} - 7\\-t^{3} + 3t\end{pmatrix}\), at \(t = 3\)

A robot vacuum cleaner moves on a floor. Its position is measured in metres east and north of the charging point at the origin. Its velocity is \(\mathbf{v} = \begin{pmatrix}3t - 4\\8e^{-2t}\end{pmatrix}\) m s⁻¹, and at \(t = 0\) it is at \(\mathbf{r}_0 = \begin{pmatrix}-1\\3\end{pmatrix}\) m. Find, when \(t = 1.5\):

1. (a) the acceleration
2. (b) the position vector
3. (c) the distance from the origin
4. (d) the speed

A model train runs on a track. Its velocity is \(\mathbf{v} = \begin{pmatrix}-15\sin 3t\\15\cos 3t\end{pmatrix}\) m s⁻¹, and at \(t = 0\) it is at \(\mathbf{r}_0 = \begin{pmatrix}5\\0\end{pmatrix}\) m. Find, when \(t = 0.9\):

1. (a) the speed
2. (b) the position vector
3. (c) the distance from the origin. What do you notice about your answers and about the shape of the path?

A drone's velocity is modelled by two components. Its velocity is \(\mathbf{v} = \begin{pmatrix}t^{2} - 6t + 4\\2t - 3\end{pmatrix}\) m s⁻¹ for \(t \ge 0\). Use your calculator to find the least speed and the time at which it happens.

A particle moves in the plane. Its velocity is \(\mathbf{v} = \begin{pmatrix}9\cos 3t\\5\sin 3t\end{pmatrix}\) m s⁻¹. Find the first time \(t > 0\) at which the particle moves parallel to \(\mathbf{j}\), and its speed then.