Velocity from acceleration and initial conditions

1. At time \(t\) seconds, where \(t \geqslant 0\), a particle \(P\) has velocity \(\mathbf { v } \mathrm { ms } ^ { - 1 }\) where

$$\mathbf { v } = \left( t ^ { 2 } - 3 t + 7 \right) \mathbf { i } + \left( 2 t ^ { 2 } - 3 \right) \mathbf { j }$$ Find

1. the speed of \(P\) at time \(t = 0\)
2. the value of \(t\) when \(P\) is moving parallel to \(( \mathbf { i } + \mathbf { j } )\)
3. the acceleration of \(P\) at time \(t\) seconds
4. the value of \(t\) when the direction of the acceleration of \(P\) is perpendicular to \(\mathbf { i }\)

1. In this question you must show all stages of your working.

\section*{Solutions relying entirely on calculator technology are not acceptable.} [In this question, \(\mathbf { i }\) is a unit vector due east and \(\mathbf { j }\) is a unit vector due north.
Position vectors are given relative to a fixed origin \(O\).] At time \(t\) seconds, \(t \geqslant 1\), the position vector of a particle \(P\) is \(\mathbf { r }\) metres, where $$\mathbf { r } = c t ^ { \frac { 1 } { 2 } } \mathbf { i } - \frac { 3 } { 8 } t ^ { 2 } \mathbf { j }$$ and \(c\) is a constant.
When \(t = 4\), the bearing of \(P\) from \(O\) is \(135 ^ { \circ }\)

1. Show that \(c = 3\)
2. Find the speed of \(P\) when \(t = 4\) When \(t = T , P\) is accelerating in the direction of ( \(\mathbf { - i } - \mathbf { 2 7 j }\) ).
3. Find the value of \(T\).

1. \hspace{0pt} [In this question \(\mathbf { i }\) and \(\mathbf { j }\) are horizontal unit vectors due east and due north respectively and position vectors are given relative to the fixed point \(O\).]

A particle \(P\) moves with constant acceleration.
At time \(t = 0\), the particle is at \(O\) and is moving with velocity ( \(2 \mathbf { i } - 3 \mathbf { j }\) ) \(\mathrm { ms } ^ { - 1 }\) At time \(t = 2\) seconds, \(P\) is at the point \(A\) with position vector ( \(7 \mathbf { i } - 10 \mathbf { j }\) ) m.

1. Show that the magnitude of the acceleration of \(P\) is \(2.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) At the instant when \(P\) leaves the point \(A\), the acceleration of \(P\) changes so that \(P\) now moves with constant acceleration ( \(4 \mathbf { i } + 8.8 \mathbf { j }\) ) \(\mathrm { m } \mathrm { s } ^ { - 2 }\) At the instant when \(P\) reaches the point \(B\), the direction of motion of \(P\) is north east.
2. Find the time it takes for \(P\) to travel from \(A\) to \(B\).

6. At time \(t\) seconds, where \(t \geqslant 0\), a particle \(P\) moves so that its acceleration \(\mathbf { a } \mathrm { m } \mathrm { s } ^ { - 2 }\) is given by $$\mathbf { a } = 5 t \mathbf { i } - 15 t ^ { \frac { 1 } { 2 } } \mathbf { j }$$ When \(t = 0\), the velocity of \(P\) is \(20 \mathbf { i } \mathrm {~m} \mathrm {~s} ^ { - 1 }\) Find the speed of \(P\) when \(t = 4\)

A particle \(P\) moves in a horizontal plane. The acceleration of \(P\) is \((-\mathbf{i} + 2\mathbf{j}) \text{ m s}^{-2}\). At time \(t = 0\), the velocity of \(P\) is \((2\mathbf{i} - 3\mathbf{j}) \text{ m s}^{-1}\).

1. Find, to the nearest degree, the angle between the vector \(\mathbf{j}\) and the direction of motion of \(P\) when \(t = 0\). [3]

At time \(t\) seconds, the velocity of \(P\) is \(\mathbf{v} \text{ m s}^{-1}\). Find

1. an expression for \(\mathbf{v}\) in terms of \(t\), in the form \(a\mathbf{i} + b\mathbf{j}\), [2]
2. the speed of \(P\) when \(t = 3\), [3]
3. the time when \(P\) is moving parallel to \(\mathbf{i}\). [2]

Fig. 4 shows the unit vectors \(\mathbf{i}\) and \(\mathbf{j}\) in the directions of the cartesian axes \(Ox\) and \(Oy\), respectively. O is the origin of the axes and of position vectors. \includegraphics{figure_1} The position vector of a particle is given by \(\mathbf{r} = 3t\mathbf{i} + (18t^2 - 11)\mathbf{j}\) for \(t \geq 0\), where \(t\) is time.

1. Show that the path of the particle cuts the \(x\)-axis just once. [2]
2. Find an expression for the velocity of the particle at time \(t\). Deduce that the particle never travels in the \(\mathbf{j}\) direction. [3]
3. Find the cartesian equation of the path of the particle, simplifying your answer. [3]

In this question, the unit vectors \(\begin{pmatrix} 1 \\ 0 \end{pmatrix}\) and \(\begin{pmatrix} 0 \\ 1 \end{pmatrix}\) are in the directions east and north. Distance is measured in metres and time, \(t\), in seconds. A radio-controlled toy car moves on a flat horizontal surface. A child is standing at the origin and controlling the car. When \(t = 0\), the displacement of the car from the origin is \(\begin{pmatrix} 0 \\ -2 \end{pmatrix}\) m, and the car has velocity \(\begin{pmatrix} 2 \\ 0 \end{pmatrix}\) ms\(^{-1}\). The acceleration of the car is constant and is \(\begin{pmatrix} -1 \\ 1 \end{pmatrix}\) ms\(^{-2}\).

1. Find the velocity of the car at time \(t\) and its speed when \(t = 8\). [4]
2. Find the distance of the car from the child when \(t = 8\). [4]

At time \(t\) seconds, a particle has position with respect to an origin O given by the vector $$\mathbf{r} = \begin{pmatrix} 8t \\ 10t^2 - 2t^3 \end{pmatrix},$$ where \(\begin{pmatrix} 1 \\ 0 \end{pmatrix}\) and \(\begin{pmatrix} 0 \\ 1 \end{pmatrix}\) are perpendicular unit vectors east and north respectively and distances are in metres.

1. When \(t = 1\), the particle is at P. Find the bearing of P from O. [2]
2. Find the velocity of the particle at time \(t\) and show that it is never zero. [3]
3. Determine the time(s), if any, when the acceleration of the particle is zero. [3]

A particle \(P\) moves with constant acceleration \((3\mathbf{i} - 5\mathbf{j})\text{m s}^{-2}\). At time \(t = 0\) seconds \(P\) is at the origin. At time \(t = 4\) seconds \(P\) has velocity \((2\mathbf{i} + 4\mathbf{j})\text{m s}^{-1}\).

1. Find the displacement vector of \(P\) at time \(t = 4\) seconds. [2]
2. Find the speed of \(P\) at time \(t = 0\) seconds. [4]