3.02a Kinematics language: position, displacement, velocity, acceleration

4. A particle \(P\) moves along the \(x\)-axis in a straight line so that, at time \(t\) seconds, the velocity of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), where $$v = \begin{cases} 10 t - 2 t ^ { 2 } , & 0 \leqslant t \leqslant 6 \\ \frac { - 432 } { t ^ { 2 } } , & t > 6 \end{cases}$$ At \(t = 0 , P\) is at the origin \(O\). Find the displacement of \(P\) from \(O\) when

1. \(t = 6\),
2. \(t = 10\).

1. A particle \(P\) moves on the \(x\)-axis. At time \(t\) seconds the velocity of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in the direction of \(x\) increasing, where \(v\) is given by

$$v = \left\{ \begin{array} { l c } 8 t - \frac { 3 } { 2 } t ^ { 2 } , & 0 \leqslant t \leqslant 4 , \\ 16 - 2 t , & t > 4 . \end{array} \right.$$ When \(t = 0 , P\) is at the origin \(O\).
Find

1. the greatest speed of \(P\) in the interval \(0 \leqslant t \leqslant 4\),
2. the distance of \(P\) from \(O\) when \(t = 4\),
3. the time at which \(P\) is instantaneously at rest for \(t > 4\),
4. the total distance travelled by \(P\) in the first 10 s of its motion.

2. At time \(t = 0\) a particle \(P\) leaves the origin \(O\) and moves along the \(x\)-axis. At time \(t\) seconds the velocity of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), where $$v = 8 t - t ^ { 2 }$$

1. Find the maximum value of \(v\).
2. Find the time taken for \(P\) to return to \(O\).

3. A particle \(P\) moves on the \(x\)-axis. At time \(t\) seconds the velocity of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in the direction of \(x\) increasing, where $$v = 2 t ^ { 2 } - 14 t + 20 , \quad t \geqslant 0$$ Find

1. the times when \(P\) is instantaneously at rest,
2. the greatest speed of \(P\) in the interval \(0 \leqslant t \leqslant 4\)
3. the total distance travelled by \(P\) in the interval \(0 \leqslant t \leqslant 4\)

1. A particle \(P\) moves on the positive \(x\)-axis. The velocity of \(P\) at time \(t\) seconds is \(\left( 2 t ^ { 2 } - 9 t + 4 \right) \mathrm { m } \mathrm { s } ^ { - 1 }\). When \(t = 0 , P\) is 15 m from the origin \(O\).

Find

1. the values of \(t\) when \(P\) is instantaneously at rest,
2. the acceleration of \(P\) when \(t = 5\)
3. the total distance travelled by \(P\) in the interval \(0 \leqslant t \leqslant 5\)

1. At time \(t = 0\) a particle \(P\) leaves the origin \(O\) and moves along the \(x\)-axis. At time \(t\) seconds, the velocity of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in the positive \(x\) direction, where

$$v = 3 t ^ { 2 } - 16 t + 21$$ The particle is instantaneously at rest when \(t = t _ { 1 }\) and when \(t = t _ { 2 } \left( t _ { 1 } < t _ { 2 } \right)\).

1. Find the value of \(t _ { 1 }\) and the value of \(t _ { 2 }\).
2. Find the magnitude of the acceleration of \(P\) at the instant when \(t = t _ { 1 }\).
3. Find the distance travelled by \(P\) in the interval \(t _ { 1 } \leqslant t \leqslant t _ { 2 }\).
4. Show that \(P\) does not return to \(O\).

1. A particle \(P\) of mass 0.5 kg moves under the action of a single force \(\mathbf { F }\) newtons. At time \(t\) seconds, \(t \geqslant 0 , P\) has velocity \(\mathbf { v } \mathrm { m } \mathrm { s } ^ { - 1 }\), where

$$\mathbf { v } = \left( 4 t - 3 t ^ { 2 } \right) \mathbf { i } + \left( t ^ { 2 } - 8 t - 40 \right) \mathbf { j }$$

1. Find
   1. the magnitude of \(\mathbf { F }\) when \(t = 3\)
   2. the acceleration of \(P\) at the instant when it is moving in the direction of the vector \(- \mathbf { i } - \mathbf { j }\). When \(t = 1 , P\) is at the point \(A\). When \(t = 2 , P\) is at the point \(B\).
2. Find, in terms of \(\mathbf { i }\) and \(\mathbf { j }\), the vector \(\overrightarrow { A B }\).

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.A particle \(P\) moves in a horizontal plane.At time \(t\) seconds,the position vector of \(P\) is \(\mathbf { r }\) metres relative to a fixed origin \(O\) ,and \(\mathbf { r }\) is given by $$\mathbf { r } = \left( 18 t - 4 t ^ { 3 } \right) \mathbf { i } + c t ^ { 2 } \mathbf { j } ,$$ where \(c\) is a positive constant.When \(t = 1.5\) ,the speed of \(P\) is \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) .Find

1. the value of \(c\) ,
2. the acceleration of \(P\) when \(t = 1.5\) . \(\mathbf { r }\) metres relative to a fixed origin \(O\) ,and \(\mathbf { r }\) is given by $$\begin{aligned} \mathbf { r } = \left( 18 t - 4 t ^ { 3 } \right) \mathbf { i } + c t ^ { 2 } \mathbf { j } , \\ \text { where } c \text { is a positive constant.When } t = 1.5 \text { ,the speed of } P \text { is } 15 \mathrm {~m} \mathrm {~s} ^ { - 1 } \text { .Find } \end{aligned}$$ (a)the value of \(c\) , 3.A particle \(P\) moves in a horizontal plane.At time \(t\) seconds,the position vector of \(P\) is D啨
   (b)the acceleration of \(P\) when \(t = 1.5\) .

1 A particle is travelling in a straight line. Its velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at time \(t\) seconds is given by $$v = 6 + 4 t \quad \text { for } 0 \leqslant t \leqslant 5$$

1. Write down the initial velocity of the particle and find the acceleration for \(0 \leqslant t \leqslant 5\).
2. Write down the velocity of the particle when \(t = 5\). Find the distance travelled in the first 5 seconds. For \(5 \leqslant t \leqslant 15\), the acceleration of the particle is \(3 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
3. Find the total distance travelled by the particle during the 15 seconds.

2 Fig. 2 shows an acceleration-time graph modelling the motion of a particle. \begin{figure}[h] \end{figure} At \(t = 0\) the particle has a velocity of \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in the positive direction.

1. Find the velocity of the particle when \(t = 2\).
2. At what time is the particle travelling in the negative direction with a speed of \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) ?

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 along a straight line. The speed of \(P\) at time \(t\) seconds ( \(t \geqslant 0\) ) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), where \(v = \left( p t ^ { 2 } + q t + r \right)\) and \(p , q\) and \(r\) are constants. When \(t = 2\) the speed of \(P\) has its minimum value. When \(t = 0 , v = 11\) and when \(t = 2 , v = 3\)

Find

1. the acceleration of \(P\) when \(t = 3\)
2. the distance travelled by \(P\) in the third second of the motion.
   

1. A particle, \(P\), moves along a straight line such that at time \(t\) seconds, \(t \geqslant 0\), the velocity of \(P\), \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), is modelled as

$$v = 12 + 4 t - t ^ { 2 }$$ Find

1. the magnitude of the acceleration of \(P\) when \(P\) is at instantaneous rest,
2. the distance travelled by \(P\) in the interval \(0 \leqslant t \leqslant 3\)

1. In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable.

A particle is moving along a straight line.
At time t seconds, \(\mathrm { t } > 0\), the velocity of the particle is \(\mathrm { Vms } ^ { - 1 }\), where $$v = 2 t - 7 \sqrt { t } + 6$$

1. Find the acceleration of the particle when \(t = 4\) When \(\mathrm { t } = 1\) the particle is at the point X .
   When \(\mathrm { t } = 2\) the particle is at the point Y . Given that the particle does not come to instantaneous rest in the interval \(1 < \mathrm { t } < 2\)
2. show that \(X Y = \frac { 1 } { 3 } ( 41 - 28 \sqrt { 2 } )\) metres.

1. A particle \(P\) moves along a straight line.

At time \(t\) seconds, the velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) of \(P\) is modelled as $$v = 10 t - t ^ { 2 } - k \quad t \geqslant 0$$ where \(k\) is a constant.

1. Find the acceleration of \(P\) at time \(t\) seconds. The particle \(P\) is instantaneously at rest when \(t = 6\)
2. Find the other value of \(t\) when \(P\) is instantaneously at rest.
3. Find the total distance travelled by \(P\) in the interval \(0 \leqslant t \leqslant 6\)

9 In this question the horizontal unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are in the directions east and north respectively.
A model ship of mass 2 kg is moving so that its acceleration vector \(\mathbf { a m s } ^ { - 2 }\) at time \(t\) seconds is given by \(\mathbf { a } = 3 ( 2 t - 5 ) \mathbf { i } + 4 \mathbf { j }\). When \(t = T\), the magnitude of the horizontal force acting on the ship is 10 N . Find the possible values of \(T\).

6 The displacement of a particle is modelled by the equation \(\mathrm { s } = 7 + 4 \mathrm { t } - \mathrm { t } ^ { 2 }\), where \(s\) metres is the displacement from the origin at time \(t\) seconds. The diagram shows part of the displacement-time graph for the particle. The point \(( 2,11 )\) is the maximum point on the graph. \includegraphics[max width=\textwidth, alt={}, center]{5428eabf-431d-4db1-8c25-1f2b9570d9aa-4_513_1381_422_255}

1. Kai argues that the point \(( 2,11 )\) is on the graph, so the particle has travelled a distance of 11 metres in the first 2 seconds. Comment on the validity of Kai's argument.
2. Determine the total distance the particle travels in the first 10 seconds.
3. Find an expression for the velocity of the particle at time \(t\).
4. Find the speed of the particle when \(t = 10\).

5 Particle P moves on a straight line that contains the point O .
At time \(t\) seconds the displacement of P from O is \(s\) metres, where \(s = t ^ { 3 } - 3 t ^ { 2 } + 3\).

1. Determine the times when the particle has zero velocity.
2. Find the distances of P from O at the times when it has zero velocity.

7 The velocity \(v \mathrm {~ms} ^ { - 1 }\) of a particle at time \(t \mathrm {~s}\) is given by \(v = 0.5 t ( 7 - t )\). Determine whether the speed of the particle is increasing or decreasing when \(t = 8\).

1 A particle moves along a straight line. The displacement \(s \mathrm {~m}\) at time \(t \mathrm {~s}\) is shown in the displacementtime graph below. The graph consists of straight line segments joining the points \(( 0 , - 2 ) , ( 10,5 )\) and \(( 15,1 )\). \includegraphics[max width=\textwidth, alt={}, center]{9dd6fc6d-b51e-4a73-ace5-d26a7558032c-04_641_848_641_242}

1. Find the distance travelled by the particle in the first 15 s .
2. Calculate the velocity of the particle between \(t = 10\) and \(t = 15\).

3 A body, \(Q\), of mass 2 kg moves in a straight line under the action of a single force which acts in the direction of motion of \(Q\). Initially the speed of \(Q\) is \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). At time \(t \mathrm {~s}\), the magnitude \(F \mathrm {~N}\) of the force is given by $$F = t ^ { 2 } + 3 \mathrm { e } ^ { t } , \quad 0 \leq t \leq 4$$

1. Calculate the impulse of the force over the time interval.
2. Hence find the speed of \(Q\) when \(t = 4\).

1. \begin{figure}[h] \end{figure} Figure 1 shows a distance-time graph for a car journey from Birmingham to Newquay which included a stop for lunch at a service station near Exeter. During the first part of the journey three-quarters of the total distance, \(d\), was covered in 3 hours. After a 1 hour stop, the remaining distance was completed in 2 hours.

1. Calculate, in the form \(k : 1\), the ratio of the average speed during the first 3 hours of the journey to the average speed during the last 2 hours of the journey.
   (4 marks)
   Given that the average speed of the car over the whole journey (excluding the stop) was \(80 \mathrm { kmh } ^ { - 1 }\),
2. find the average speed of the car on the first part of the journey.
   (4 marks)

4. In this question, \(\mathbf { i }\) and \(\mathbf { j }\) are perpendicular horizontal unit vectors and \(O\) is a fixed origin. A pedestrian moves with constant velocity \(\left[ \left( 2 q ^ { 2 } - 3 \right) \mathbf { i } + ( q + 2 ) \mathbf { j } \right] \mathrm { ms } ^ { - 1 }\). Given that the velocity of the pedestrian is parallel to the vector \(( \mathbf { i } - \mathbf { j } )\),

1. Show that one possible value of \(q\) is \({ } ^ { - } 1\) and find the other possible value of \(q\). Given that \(q = { } ^ { - } 1\), and that the pedestrian started walking at the point with position vector \(( 6 \mathbf { i } - \mathbf { j } ) \mathrm { m }\),
2. find the length of time for which the pedestrian is less than 5 m from \(O\).