Finding when moving in specific direction

1. A particle \(P\) of mass 2 kg is moving under the action of a constant force \(\mathbf { F }\) newtons. The velocity of \(P\) is \(( 2 \mathbf { i } - 5 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) at time \(t = 0\), and \(( 7 \mathbf { i } + 10 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) at time \(t = 5 \mathrm {~s}\).

Find

1. the speed of \(P\) at \(t = 0\),
2. the vector \(\mathbf { F }\) in the form \(a \mathbf { i } + b \mathbf { j }\),
3. the value of \(t\) when \(P\) is moving parallel to \(\mathbf { i }\).

1. \hspace{0pt} [In this question, the horizontal unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are directed due east and due north respectively.]

The velocity, \(\mathbf { v } \mathrm { m } \mathrm { s } ^ { - 1 }\), of a particle \(P\) at time \(t\) seconds is given by $$\mathbf { v } = ( 1 - 2 t ) \mathbf { i } + ( 3 t - 3 ) \mathbf { j }$$

1. Find the speed of \(P\) when \(t = 0\)
2. Find the bearing on which \(P\) is moving when \(t = 2\)
3. Find the value of \(t\) when \(P\) is moving
   1. parallel to \(\mathbf { j }\),
   2. parallel to \(( - \mathbf { i } - 3 \mathbf { j } )\).

1. At time \(t\) seconds \(( t \geqslant 0 )\) a particle \(P\) has position vector \(\mathbf { r }\) metres, with respect to a fixed origin \(O\), where

$$\mathbf { r } = \left( \frac { 1 } { 8 } t ^ { 4 } - 2 \lambda t ^ { 2 } + 5 \right) \mathbf { i } + \left( 5 t ^ { 2 } - \lambda t \right) \mathbf { j }$$ and \(\lambda\) is a constant. When \(t = 4 , P\) is moving parallel to the vector \(\mathbf { j }\).

1. Show that \(\lambda = 2\)
2. Find the speed of \(P\) when \(t = 4\)
3. Find the acceleration of \(P\) when \(t = 4\) When \(t = 0 , P\) is at the point \(A\). When \(t = 4 , P\) is at the point \(B\).
4. Find the distance \(A B\).

5 The position vector of a particle at time \(t\) is given by $$\mathbf { r } = \frac { 1 } { 2 } t \mathbf { i } + \left( t ^ { 2 } - 1 \right) \mathbf { j } ,$$ referred to an origin \(\mathbf { O }\) where \(\mathbf { i }\) and \(\mathbf { j }\) are the standard unit vectors in the directions of the cartesian axes \(\mathrm { O } x\) and Oy respectively.

1. Write down the value of \(t\) for which the \(x\)-coordinate of the position of the particle is 2 . Find the \(y\)-coordinate at this time.
2. Show that the cartesian equation of the path of the particle is \(y = 4 x ^ { 2 } - 1\).
3. Find the coordinates of the point where the particle is moving at \(45 ^ { \circ }\) to both \(\mathrm { O } x\) and \(\mathrm { O } y\). Section B (36 marks)

1. At time \(t\) seconds, \(t \geqslant 0\), a particle \(P\) has position vector \(\mathbf { r }\) metres with respect to a fixed origin \(O\), where

$$\mathbf { r } = \left( t ^ { 3 } - 8 t \right) \mathbf { i } + \left( \frac { 1 } { 3 } t ^ { 3 } - t ^ { 2 } + 2 t \right) \mathbf { j }$$

1. Find the acceleration of \(P\) when \(t = 4\) At time \(T\) seconds, \(T \geqslant 0 , P\) is moving in the direction of ( \(2 \mathbf { i } + \mathbf { j }\) )
2. Find the value of \(T\)

1. At time \(t\) seconds \(( t \geqslant 0 )\), a particle \(P\) has position vector \(\mathbf { r }\) metres with respect to a fixed origin \(O\), where

$$\mathbf { r } = \left( t ^ { 3 } - \frac { 9 } { 2 } t ^ { 2 } - 24 t \right) \mathbf { i } + \left( - t ^ { 3 } + 3 t ^ { 2 } + 12 t \right) \mathbf { j }$$ At time \(T\) seconds, \(P\) is moving in a direction parallel to the vector \(\mathbf { - i } - \mathbf { j }\)
Find

1. the value of \(T\),
2. the magnitude of the acceleration of \(P\) at the instant when \(t = T\).

3. At time \(t\) seconds \(( t \geqslant 0 )\) a particle \(P\) has position vector \(\mathbf { r }\) metres, with respect to a fixed origin \(O\), where
(b) the magnitude of the acceleration of \(P\) when \(t = 4\) $$\begin{aligned} & \qquad \mathbf { r } = \left( 16 t - 3 t ^ { 3 } \right) \mathbf { i } + \left( t ^ { 3 } - t ^ { 2 } + 2 \right) \mathbf { j } \\ & \text { Find } \\ & \text { (a) the velocity of } P \text { at the instant when it is moving parallel to the vector } \mathbf { j } \text {, } \end{aligned}$$ VILIV SIHI NI IIIIIM ION OC
VILV SIHI NI JAHAM ION OC
VJ4V SIHI NI JIIYM ION OC

1. At time \(t\) seconds, \(t > 0\), a particle \(P\) is at the point with position vector \(\mathbf { r } \mathrm { m }\), where

$$\mathbf { r } = \left( t ^ { 4 } - 8 t ^ { 2 } \right) \mathbf { i } + \left( 6 t ^ { 2 } - 2 t ^ { \frac { 3 } { 2 } } \right) \mathbf { j }$$

1. Find the velocity of \(P\) when \(P\) is moving in a direction parallel to the vector \(\mathbf { j }\)
2. Find the acceleration of \(P\) when \(t = 4\)

3. At time \(t\) seconds \(( t \geqslant 0 )\) a particle \(P\) has velocity \(\mathbf { v } \mathrm { m } \mathrm { s } ^ { - 1 }\), where $$\mathbf { v } = \left( 6 t ^ { 2 } + 6 t \right) \mathbf { i } + \left( 3 t ^ { 2 } + 24 \right) \mathbf { j }$$ When \(t = 0\) the particle \(P\) is at the origin \(O\). At time \(T\) seconds, \(P\) is at the point \(A\) and \(\mathbf { v } = \lambda ( \mathbf { i } + \mathbf { j } )\), where \(\lambda\) is a constant. Find

1. the value of \(T\),
2. the acceleration of \(P\) as it passes through the point \(A\),
3. the distance \(O A\).
   

3. At time \(t\) seconds, a particle \(P\) has position vector \(\mathbf { r }\) metres relative to a fixed origin \(O\), where $$\mathbf { r } = \left( t ^ { 3 } - 3 t \right) \mathbf { i } + 4 t ^ { 2 } \mathbf { j } , t \geq 0$$ Find

1. the velocity of \(P\) at time \(t\) seconds,
2. the time when \(P\) is moving parallel to the vector \(\mathbf { i } + \mathbf { j }\).
   (5)

3．At time \(t\) seconds（ \(t \geqslant 0\) ）a particle \(P\) has velocity \(\mathbf { v } \mathrm { ms } ^ { - 1 }\) ，where When \(t = 0\) the particle \(P\) is at the origin \(O\) ．At time \(T\) seconds，\(P\) is at the point \(A\) and \(\mathbf { v } = \lambda ( \mathbf { i } + \mathbf { j } )\) ，where \(\lambda\) is a constant． Find
（a）the value of \(T\) ，
（b）the acceleration of \(P\) as it passes through the point \(A\) ，
（c）the distance \(O A\) ． $$\mathbf { v } = \left( 6 t ^ { 2 } + 6 t \right) \mathbf { i } + \left( 3 t ^ { 2 } + 24 \right) \mathbf { j }$$ 的 When \(t = 0\) the particle \(P\) is at the origin \(O\) ．At time \(T\) seconds，\(P\) is at the point \(A\) and
\(\mathbf { v } = \lambda ( \mathbf { i } + \mathbf { j } )\) ，where \(\lambda\) is a constant． Find
（a）the value of \(T\) ，
\(\_\_\_\_\) ＂

8 A toy boat moves in a horizontal plane with position vector \(\mathbf { r } = x \mathbf { i } + y \mathbf { j }\), where \(\mathbf { i }\) and \(\mathbf { j }\) are the standard unit vectors east and north respectively. The origin of the position vectors is at O . The displacements \(x\) and \(y\) are in metres. First consider only the motion of the boat parallel to the \(x\)-axis. For this motion $$x = 8 t - 2 t ^ { 2 }$$ The velocity of the boat in the \(x\)-direction is \(v _ { x } \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Find an expression in terms of \(t\) for \(v _ { x }\) and determine when the boat instantaneously has zero speed in the \(x\)-direction. Now consider only the motion of the boat parallel to the \(y\)-axis. For this motion $$v _ { y } = ( t - 2 ) ( 3 t - 2 )$$ where \(v _ { y } \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the velocity of the boat in the \(y\)-direction at time \(t\) seconds.
2. Given that \(y = 3\) when \(t = 1\), use integration to show that \(y = t ^ { 3 } - 4 t ^ { 2 } + 4 t + 2\). The position vector of the boat is given in terms of \(t\) by \(\mathbf { r } = \left( 8 t - 2 t ^ { 2 } \right) \mathbf { i } + \left( t ^ { 3 } - 4 t ^ { 2 } + 4 t + 2 \right) \mathbf { j }\).
3. Find the time(s) when the boat is due north of O and also the distance of the boat from O at any such times.
4. Find the time(s) when the boat is instantaneously at rest. Find the distance of the boat from O at any such times.
5. Plot a graph of the path of the boat for \(0 \leqslant t \leqslant 2\).

4 The directions of the unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are east and north.
The velocity of a particle, \(\mathrm { vm } \mathrm { s } ^ { - 1 }\), at time \(t \mathrm {~s}\) is given by $$\mathbf { v } = \left( 16 - t ^ { 2 } \right) \mathbf { i } + ( 31 - 8 t ) \mathbf { j } .$$ Find the time at which the particle is travelling on a bearing of \(045 ^ { \circ }\) and the speed of the particle at this time.

1. A particle, \(P\), moves with constant acceleration \(( 2 \mathbf { i } - 3 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 2 }\)

At time \(t = 0\), the particle is at the point \(A\) and is moving with velocity ( \(- \mathbf { i } + 4 \mathbf { j }\) ) \(\mathrm { m } \mathrm { s } ^ { - 1 }\)
At time \(t = T\) seconds, \(P\) is moving in the direction of vector ( \(3 \mathbf { i } - 4 \mathbf { j }\) )

1. Find the value of \(T\). At time \(t = 4\) seconds, \(P\) is at the point \(B\).
2. Find the distance \(A B\).

1. 1. At time \(t\) seconds, where \(t \geqslant 0\), a particle \(P\) moves so that its acceleration a \(\mathrm { ms } ^ { - 2 }\) is given by

$$\mathbf { a } = ( 1 - 4 t ) \mathbf { i } + \left( 3 - t ^ { 2 } \right) \mathbf { j }$$ At the instant when \(t = 0\), the velocity of \(P\) is \(36 \mathbf { i } \mathrm {~m} \mathrm {~s} ^ { - 1 }\)

1. Find the velocity of \(P\) when \(t = 4\)
2. Find the value of \(t\) at the instant when \(P\) is moving in a direction perpendicular to i
   (ii) At time \(t\) seconds, where \(t \geqslant 0\), a particle \(Q\) moves so that its position vector \(\mathbf { r }\) metres, relative to a fixed origin \(O\), is given by $$\mathbf { r } = \left( t ^ { 2 } - t \right) \mathbf { i } + 3 t \mathbf { j }$$ Find the value of \(t\) at the instant when the speed of \(Q\) is \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\)

12 A model boat has velocity \(\mathbf { v } = ( ( 2 t - 2 ) \mathbf { i } + ( 2 t + 2 ) \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) for \(t \geq 0\), where \(t\) is the time in seconds. \(\mathbf { i }\) is the unit vector east and \(\mathbf { j }\) is the unit vector north.
When \(t = 3\), the position vector of the boat is \(( 3 \mathbf { i } + 14 \mathbf { j } ) \mathrm { m }\).

1. Show that the boat is never instantaneously at rest.
2. Determine any times at which the boat is moving directly northwards.
3. Determine any times at which the boat is north-east of the origin.

2 The directions of the unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are east and north.
The velocity of a particle, \(\mathbf { v } \mathrm { m } \mathrm { s } ^ { - 1 }\), at time \(t \mathrm {~s}\) is given by $$\mathbf { v } = \left( 16 - t ^ { 2 } \right) \mathbf { i } + ( 31 - 8 t ) \mathbf { j }$$ Find the time at which the particle is travelling on a bearing of \(045 ^ { \circ }\) and the speed of the particle at this time.
[0pt] [6]

6 A toy boat moves in a horizontal plane with position vector \(\mathbf { r } = x \mathbf { i } + y \mathbf { j }\), where \(\mathbf { i }\) and \(\mathbf { j }\) are the standard unit vectors east and north respectively. The origin of the position vectors is at O . The displacements \(x\) and \(y\) are in metres. First consider only the motion of the boat parallel to the \(x\)-axis. For this motion $$x = 8 t - 2 t ^ { 2 }$$ The velocity of the boat in the \(x\)-direction is \(v _ { x } \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Find an expression in terms of \(t\) for \(v _ { x }\) and determine when the boat instantaneously has zero speed in the \(x\)-direction. Now consider only the motion of the boat parallel to the \(y\)-axis. For this motion $$v _ { y } = ( t - 2 ) ( 3 t - 2 )$$ where \(v _ { y } \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the velocity of the boat in the \(y\)-direction at time \(t\) seconds.
2. Given that \(y = 3\) when \(t = 1\), use integration to show that \(y = t ^ { 3 } - 4 t ^ { 2 } + 4 t + 2\). The position vector of the boat is given in terms of \(t\) by \(\mathbf { r } = \left( 8 t - 2 t ^ { 2 } \right) \mathbf { i } + \left( t ^ { 3 } - 4 t ^ { 2 } + 4 t + 2 \right) \mathbf { j }\).
3. Find the time(s) when the boat is due north of O and also the distance of the boat from O at any such times.
4. Find the time(s) when the boat is instantaneously at rest. Find the distance of the boat from O at any such times.
5. Plot a graph of the path of the boat for \(0 \leqslant t \leqslant 2\).

2 The position vector of a particle at time \(t\) is given by $$\mathbf { r } = \frac { 1 } { 2 } t \mathbf { i } + \left( t ^ { 2 } - 1 \right) \mathbf { j } .$$ referred to an origin \(O\) where \(\mathbf { i }\) and \(\mathbf { j }\) are the standard unit vectors in the directions of the cartesian axes Ox and Oy respectively.

1. Write down the value of \(t\) for which the \(x\)-coordinate of the position of the particle is 2 . Find the \(y\)-coordinate at this time.
2. Show that the cartesian equation of the path of the particle is \(y = 4 x ^ { 2 } - 1\).
3. Find the coordinates of the point where the particle is moving at \(45 ^ { \circ }\) to both Ox and Oy .

2 A particle, Q , moves so that its velocity, \(\mathbf { v }\), at time \(t\) is given by \(\mathbf { v } = ( 6 t - 6 ) \mathbf { i } + \left( 3 - 2 t + t ^ { 2 } \right) \mathbf { j } + 4 \mathbf { k }\), where \(0 \leqslant t \leqslant 6\).

1. Explain how you know that Q is never stationary. When Q is at a point A the direction of the acceleration of Q is parallel to the \(\mathbf { i }\) direction. When Q is at a point B the direction of the acceleration of Q makes an angle of \(45 ^ { \circ }\) with the \(\mathbf { i }\) direction.
2. Determine the straight-line distance AB .

6 At time \(t\) seconds, where \(t \geqslant 0\), a particle P has position vector \(\mathbf { r }\) metres, where
\(\mathbf { r } = \left( 2 t ^ { 2 } - 12 t + 6 \right) \mathbf { i } + \left( t ^ { 3 } + 3 t ^ { 2 } - 8 t \right) \mathbf { j }\).
The velocity of P at time \(t\) seconds is \(\mathbf { v } \mathrm { ms } ^ { - 1 }\).

1. Find \(\mathbf { v }\) in terms of \(t\).
2. Determine the speed of P at the instant when it is moving parallel to the vector \(\mathbf { i } - 4 \mathbf { j }\).
3. Determine the value of \(t\) when the magnitude of the acceleration of P is \(20.2 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).

4 A particle P moves so that its position vector \(\mathbf { r }\) at time \(t\) is given by \(\mathbf { r } = ( 5 + 20 t ) \mathbf { i } + \left( 95 + 10 t - 5 t ^ { 2 } \right) \mathbf { j }\).

1. Determine the initial velocity of P . At time \(t = T , \mathrm { P }\) is moving in a direction perpendicular to its initial direction of motion.
2. Determine the value of \(T\).
3. Determine the distance of P from its initial position at time \(T\).

5. A particle of mass 2 kg is moving under the action of a force \(\mathbf { F N }\) which, at time \(t\) seconds, is given by $$\mathbf { F } = 4 t \mathbf { i } - \sqrt { t } \mathbf { j } + 6 \mathbf { k }$$ When \(t = 1\), the velocity of the particle is \(\left( 3 \mathbf { i } - \frac { 1 } { 3 } \mathbf { j } - \mathbf { k } \right) \mathrm { ms } ^ { - 1 }\).

1. Find an expression for the velocity vector of the particle at time \(t \mathrm {~s}\).
2. Determine the values of \(t\) when the particle is moving in a direction perpendicular to the vector \(( - \mathbf { i } + 3 \mathbf { k } )\).

8 At time \(t\) seconds a particle \(P\) has position vector \(\mathbf { r }\) metres, with respect to a fixed origin \(O\), where $$\mathbf { r } = \left( 4 t ^ { 2 } - k t + 5 \right) \mathbf { i } + \left( 4 t ^ { 3 } + 2 k t ^ { 2 } - 8 t \right) \mathbf { j } , \quad t \geqslant 0 .$$ When \(t = 2 , P\) is moving parallel to the vector \(\mathbf { i }\).

1. Show that \(k = - 5\).
2. Find the values of \(t\) when the magnitude of the acceleration of \(P\) is \(10 \mathrm {~ms} ^ { - 2 }\).