Verifying given motion properties

6 A particle starts from a point \(O\) and moves in a straight line. The velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) of the particle at time \(t \mathrm {~s}\) after leaving \(O\) is given by $$v = k \left( 3 t ^ { 2 } - 2 t ^ { 3 } \right)$$ where \(k\) is a constant.

1. Verify that the particle returns to \(O\) when \(t = 2\).
2. It is given that the acceleration of the particle is \(- 13.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) for the positive value of \(t\) at which \(v = 0\). Find \(k\) and hence find the total distance travelled in the first two seconds of motion.

7 A particle \(P\) moves on a straight line. It starts at a point \(O\) on the line and returns to \(O 100 \mathrm {~s}\) later. The velocity of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at time \(t \mathrm {~s}\) after leaving \(O\), where $$v = 0.0001 t ^ { 3 } - 0.015 t ^ { 2 } + 0.5 t$$

1. Show that \(P\) is instantaneously at rest when \(t = 0 , t = 50\) and \(t = 100\).
2. Find the values of \(v\) at the times for which the acceleration of \(P\) is zero, and sketch the velocitytime graph for \(P\) 's motion for \(0 \leqslant t \leqslant 100\).
3. Find the greatest distance of \(P\) from \(O\) for \(0 \leqslant t \leqslant 100\). \footnotetext{Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonable effort has been made by the publisher (UCLES) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher will be pleased to make amends at the earliest possible opportunity.
   To avoid the issue of disclosure of answer-related information to candidates, all copyright acknowledgements are reproduced online in the Cambridge International Examinations Copyright Acknowledgements Booklet. This is produced for each series of examinations and is freely available to download at \href{http://www.cie.org.uk}{www.cie.org.uk} after the live examination series. Cambridge International Examinations is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of University of Cambridge Local Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge. }

7 A particle \(P\) starts from a point \(O\) and moves along a straight line. \(P\) 's velocity \(t\) s after leaving \(O\) is \(\nu \mathrm { m } \mathrm { s } ^ { - 1 }\), where $$v = 0.16 t ^ { \frac { 3 } { 2 } } - 0.016 t ^ { 2 } .$$ \(P\) comes to rest instantaneously at the point \(A\).

1. Verify that the value of \(t\) when \(P\) is at \(A\) is 100 .
2. Find the maximum speed of \(P\) in the interval \(0 < t < 100\).
3. Find the distance \(O A\).
4. Find the value of \(t\) when \(P\) passes through \(O\) on returning from \(A\).

6 A particle \(P\) starts from rest at a point \(O\) of a straight line and moves along the line. The displacement of the particle at time \(t \mathrm {~s}\) after leaving \(O\) is \(x \mathrm {~m}\), where $$x = 0.08 t ^ { 2 } - 0.0002 t ^ { 3 }$$

1. Find the value of \(t\) when \(P\) returns to \(O\) and find the speed of \(P\) as it passes through \(O\) on its return.
2. For the motion of \(P\) until the instant it returns to \(O\), find
   (a) the total distance travelled,
   (b) the average speed.

5 A particle starts from a fixed origin with velocity \(0.4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and moves in a straight line. The acceleration \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\) of the particle \(t \mathrm {~s}\) after it leaves the origin is given by \(a = k \left( 3 t ^ { 2 } - 12 t + 2 \right)\), where \(k\) is a constant. When \(t = 1\), the velocity of \(P\) is \(0.1 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Show that the value of \(k\) is 0.1 .
2. Find an expression for the displacement of the particle from the origin in terms of \(t\).
3. Hence verify that the particle is again at the origin at \(t = 2\).

4 The displacement of a particle from a fixed point \(O\) at time \(t\) seconds is \(t ^ { 4 } - 8 t ^ { 2 } + 16\) metres, where \(t \geqslant 0\).

1. Verify that when \(t = 2\) the particle is at rest at the point \(O\).
2. Calculate the acceleration of the particle when \(t = 2\).

2 A particle \(P\) moves in a straight line. The displacement of \(P\) from a fixed point on the line is \(\left( t ^ { 4 } - 2 t ^ { 3 } + 5 \right) \mathrm { m }\), where \(t\) is the time in seconds. Show that, when \(t = 1.5\),

1. \(P\) is at instantaneous rest,
2. the acceleration of \(P\) is \(9 \mathrm {~ms} ^ { - 2 }\).

3 A particle has mass 800 kg . A single force of \(( 2400 \mathbf { i } - 4800 t \mathbf { j } )\) newtons acts on the particle at time \(t\) seconds. No other forces act on the particle.

1. Find the acceleration of the particle at time \(t\).
2. At time \(t = 0\), the velocity of the particle is \(( 6 \mathbf { i } + 30 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\). The velocity of the particle at time \(t\) is \(\mathbf { v } \mathrm { m } \mathrm { s } ^ { - 1 }\). Show that $$\mathbf { v } = ( 6 + 3 t ) \mathbf { i } + \left( 30 - 3 t ^ { 2 } \right) \mathbf { j }$$
3. Initially, the particle is at the point with position vector \(( 2 \mathbf { i } + 5 \mathbf { j } ) \mathrm { m }\). Find the position vector, \(\mathbf { r }\) metres, of the particle at time \(t\).

4. The velocity \(\mathbf { v } \mathrm { ms } ^ { - 1 }\) of a particle \(P\) at time \(t\) seconds is given by \(\mathbf { v } = 3 t \mathbf { i } - t ^ { 2 } \mathbf { j }\).

1. Find the magnitude of the acceleration of \(P\) when \(t = 2\). When \(t = 0\), the displacement of \(P\) from a fixed origin \(O\) is \(( 6 \mathbf { i } + 12 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\), where \(\mathbf { i }\) and \(\mathbf { j }\) are perpendicular horizontal unit vectors.
2. Show that the displacement of \(P\) from \(O\) when \(t = 6\) is given by \(k ( \mathbf { i } - \mathbf { j } ) \mathrm { m }\), where \(k\) is an integer which you should find.
   (6 marks)

2 A particle P of mass \(m\) travels in a straight line on a smooth horizontal surface.
At time \(t , \mathrm { P }\) is a distance \(x\) from a fixed point O and is moving with speed \(v\) away from O . A horizontal force of magnitude \(3 m t\) acts on P , in a direction away from O .

1. Show that \(\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } = 3 t\).
2. Verify that the general solution of this differential equation is \(x = \frac { 1 } { 2 } t ^ { 3 } + A t + k\), where \(A\) and \(k\) are constants.
3. Given that \(x = 6\) and \(v = 12\) when \(t = 1\), find the values of \(A\) and \(k\).