3.02f Non-uniform acceleration: using differentiation and integration

375 questions

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AQA M2 2012 January Q2
10 marks Standard +0.3
2 A particle, of mass 50 kg , moves on a smooth horizontal plane. A single horizontal force $$\left[ \left( 300 t - 60 t ^ { 2 } \right) \mathbf { i } + 100 \mathrm { e } ^ { - 2 t } \mathbf { j } \right] \text { newtons }$$ acts on the particle at time \(t\) seconds.
The vectors \(\mathbf { i }\) and \(\mathbf { j }\) are perpendicular unit vectors.
  1. Find the acceleration of the particle at time \(t\).
  2. When \(t = 0\), the velocity of the particle is \(( 7 \mathbf { i } - 4 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\). Find the velocity of the particle at time \(t\).
  3. Calculate the speed of the particle when \(t = 1\).
AQA M2 2012 January Q6
10 marks Standard +0.3
6 Alice places a toy, of mass 0.4 kg , on a slope. The toy is set in motion with an initial velocity of \(1 \mathrm {~ms} ^ { - 1 }\) down the slope. The resultant force acting on the toy is \(( 2 - 4 v )\) newtons, where \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the toy's velocity at time \(t\) seconds after it is set in motion.
  1. Show that \(\frac { \mathrm { d } v } { \mathrm {~d} t } = - 10 ( v - 0.5 )\).
  2. By using \(\int \frac { 1 } { v - 0.5 } \mathrm {~d} v = - \int 10 \mathrm {~d} t\), find \(v\) in terms of \(t\).
  3. Find the time taken for the toy's velocity to reduce to \(0.55 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). \(7 \quad\) A small bead, of mass \(m\), is suspended from a fixed point \(O\) by a light inextensible string of length \(a\). With the string taut, the bead is at the point \(B\), vertically below \(O\), when it is set into vertical circular motion with an initial horizontal velocity \(u\), as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{06c3e260-8167-4616-97d4-0f360a376a0f-5_616_613_520_733} The string does not become slack in the subsequent motion. The velocity of the bead at the point \(A\), where \(A\) is vertically above \(O\), is \(v\).
AQA M2 2013 January Q2
11 marks Standard +0.3
2 A particle moves in a horizontal plane. The vectors \(\mathbf { i }\) and \(\mathbf { j }\) are perpendicular unit vectors in the horizontal plane. At time \(t\) seconds, the velocity of the particle, \(\mathbf { v } \mathrm { m } \mathrm { s } ^ { - 1 }\), is given by $$\mathbf { v } = 12 \cos \left( \frac { \pi } { 3 } t \right) \mathbf { i } - 9 t ^ { 2 } \mathbf { j }$$
  1. Find an expression for the acceleration of the particle at time \(t\).
  2. The particle, which has mass 4 kg , moves under the action of a single force, \(\mathbf { F }\) newtons.
    1. Find an expression for the force \(\mathbf { F }\) in terms of \(t\).
    2. Find the magnitude of \(\mathbf { F }\) when \(t = 3\).
  3. When \(t = 3\), the particle is at the point with position vector \(( 4 \mathbf { i } - 2 \mathbf { j } ) \mathrm { m }\). Find the position vector, \(\mathbf { r }\) metres, of the particle at time \(t\).
AQA M2 2006 June Q1
12 marks Moderate -0.8
1 A particle moves in a horizontal plane, in which the unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are directed east and north respectively. At time \(t\) seconds, its position vector, \(\mathbf { r }\) metres, is given by $$\mathbf { r } = \left( 2 t ^ { 3 } - t ^ { 2 } + 6 \right) \mathbf { i } + \left( 8 - 4 t ^ { 3 } + t \right) \mathbf { j }$$
  1. Find an expression for the velocity of the particle at time \(t\).
    1. Find the velocity of the particle when \(t = \frac { 1 } { 3 }\).
    2. State the direction in which the particle is travelling at this time.
  3. Find the acceleration of the particle when \(t = 4\).
  4. The mass of the particle is 6 kg . Find the magnitude of the resultant force on the particle when \(t = 4\).
AQA M2 2006 June Q5
14 marks Standard +0.3
5 The graph shows a model for the resultant horizontal force on a car, which varies as it accelerates from rest for 20 seconds. The mass of the car is 1200 kg . \includegraphics[max width=\textwidth, alt={}, center]{c02cf013-365b-44e2-8c16-aa8209cbe250-4_373_1203_445_390}
  1. The acceleration of the car at time \(t\) seconds is \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\). Show that $$a = \frac { 2 } { 3 } + \frac { t } { 20 } , \text { for } 0 \leqslant t \leqslant 20$$
  2. Find an expression for the velocity of the car at time \(t\).
  3. Find the distance travelled by the car in the 20 seconds.
  4. An alternative model assumes that the resultant force increases uniformly from 900 to 2100 newtons during the 20 seconds. Which term in your expression for the velocity would change as a result of this modification? Explain why.
AQA M2 2007 June Q3
11 marks Moderate -0.3
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 \(t\).
AQA M2 2010 June Q4
12 marks Standard +0.3
4 A particle has mass 200 kg and moves on a smooth horizontal plane. A single horizontal force, \(\left( 400 \cos \left( \frac { \pi } { 2 } t \right) \mathbf { i } + 600 t ^ { 2 } \mathbf { j } \right)\) newtons, acts on the particle at time \(t\) seconds. The unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are directed east and north respectively.
  1. Find the acceleration of the particle at time \(t\).
  2. When \(t = 4\), the velocity of the particle is \(( - 3 \mathbf { i } + 56 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\). Find the velocity of the particle at time \(t\).
  3. Find \(t\) when the particle is moving due west.
  4. Find the speed of the particle when it is moving due west.
    \includegraphics[max width=\textwidth, alt={}]{3ffa0a2b-aa7d-46eb-b92b-3e3ee59f235c-09_2484_1709_223_153}
AQA M2 2011 June Q3
14 marks Standard +0.3
3 A particle moves in a horizontal plane under the action of a single force, \(\mathbf { F }\) newtons. The unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are directed east and north respectively. At time \(t\) seconds, the velocity of the particle, \(\mathbf { v } \mathrm { ms } ^ { - 1 }\), is given by $$\mathbf { v } = 4 \mathrm { e } ^ { - 2 t } \mathbf { i } + \left( 6 t - 3 t ^ { 2 } \right) \mathbf { j }$$
  1. Find an expression for the acceleration of the particle at time \(t\).
  2. The mass of the particle is 5 kg .
    1. Find an expression for the force \(\mathbf { F }\) acting on the particle at time \(t\).
    2. Find the magnitude of \(\mathbf { F }\) when \(t = 0\).
  3. Find the value of \(t\) when \(\mathbf { F }\) acts due west.
  4. When \(t = 0\), the particle is at the point with position vector \(( 6 \mathbf { i } + 5 \mathbf { j } ) \mathrm { m }\). Find the position vector, \(\mathbf { r }\) metres, of the particle at time \(t\).
AQA M2 2012 June Q2
9 marks Moderate -0.3
2 A particle moves in a straight line. At time \(t\) seconds, it has velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), where $$v = 6 t ^ { 2 } - 2 \mathrm { e } ^ { - 4 t } + 8$$ and \(t \geqslant 0\).
    1. Find an expression for the acceleration of the particle at time \(t\).
    2. Find the acceleration of the particle when \(t = 0.5\).
  2. The particle has mass 4 kg . Find the magnitude of the force acting on the particle when \(t = 0.5\).
  3. When \(t = 0\), the particle is at the origin. Find an expression for the displacement of the particle from the origin at time \(t\).
AQA M2 2012 June Q4
9 marks Moderate -0.3
4 A particle moves on a horizontal plane, in which the unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are perpendicular. At time \(t\), the particle's position vector, \(\mathbf { r }\), is given by $$\mathbf { r } = 4 \cos 3 t \mathbf { i } - 4 \sin 3 t \mathbf { j }$$
  1. Prove that the particle is moving on a circle, which has its centre at the origin.
  2. Find an expression for the velocity of the particle at time \(t\).
  3. Find an expression for the acceleration of the particle at time \(t\).
  4. The acceleration of the particle can be written as $$\mathbf { a } = k \mathbf { r }$$ where \(k\) is a constant. Find the value of \(k\).
  5. State the direction of the acceleration of the particle.
AQA M2 2013 June Q1
6 marks Easy -1.2
1 A particle, of mass 3 kg , moves along a straight line. At time \(t\) seconds, the displacement, \(s\) metres, of the particle from the origin is given by $$s = 8 t ^ { 3 } + 15$$
  1. Find the velocity of the particle at time \(t\).
  2. Find the magnitude of the resultant force acting on the particle when \(t = 2\).
AQA M2 2013 June Q3
8 marks Standard +0.3
3 A particle, of mass 10 kg , moves on a smooth horizontal plane. At time \(t\) seconds, the acceleration of the particle is given by $$\left\{ \left( 40 t + 3 t ^ { 2 } \right) \mathbf { i } + 20 \mathrm { e } ^ { - 4 t } \mathbf { j } \right\} \mathrm { m } \mathrm {~s} ^ { - 2 }$$ where the vectors \(\mathbf { i }\) and \(\mathbf { j }\) are perpendicular unit vectors.
  1. At time \(t = 1\), the velocity of the particle is \(\left( 6 \mathbf { i } - 5 \mathrm { e } ^ { - 4 } \mathbf { j } \right) \mathrm { m } \mathrm { s } ^ { - 1 }\). Find the velocity of the particle at time \(t\).
  2. Calculate the initial speed of the particle.
AQA M2 2015 June Q1
10 marks Standard +0.3
1 A particle, of mass 4 kg , moves in a horizontal plane under the action of a single force, \(\mathbf { F }\) newtons. The unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are in the horizontal plane, perpendicular to each other. At time \(t\) seconds, the velocity of the particle, \(\mathbf { v } \mathrm { m } \mathrm { s } ^ { - 1 }\), is given by $$\mathbf { v } = 4 \cos 2 t \mathbf { i } + 3 \sin t \mathbf { j }$$
    1. Find an expression for the force, \(\mathbf { F }\), acting on the particle at time \(t\) seconds.
    2. Find the magnitude of \(\mathbf { F }\) when \(t = \pi\).
  2. When \(t = 0\), the particle is at the point with position vector \(( 2 \mathbf { i } - 14 \mathbf { j } )\) metres. Find the position vector, \(\mathbf { r }\) metres, of the particle at time \(t\) seconds.
    [0pt] [5 marks]
    \includegraphics[max width=\textwidth, alt={}]{691c50b4-50b2-4e3a-a7e0-60f8ec35ee3c-02_1346_1717_1361_150}
Edexcel M2 Q1
4 marks Standard +0.2
  1. A car of mass 1200 kg decelerates from \(30 \mathrm {~ms} ^ { - 1 }\) to \(20 \mathrm {~ms} ^ { - 1 }\) in 6 seconds at a constant rate.
    1. Find the magnitude, in N , of the decelerating force.
    2. Find the loss, in J , in the car's kinetic energy.
    3. A particle moves in a straight line from \(A\) to \(B\) in 5 seconds. At time \(t\) seconds after leaving \(A\), the velocity of the particle is \(\left( 32 t - 3 t ^ { 2 } \right) \mathrm { ms } ^ { - 1 }\).
    1. Calculate the straight-line distance \(A B\).
    2. Find the acceleration of the particle when \(t = 3\).
    3. Eddie, whose mass is 71 kg , rides a bicycle of mass 25 kg up a hill inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac { 1 } { 12 }\). When Eddie is working at a rate of 600 W , he is moving at a constant speed of \(6 \mathrm {~ms} ^ { - 1 }\).
      Find the magnitude of the non-gravitational resistance to his motion.
    4. A boat leaves the point \(O\) and moves such that, \(t\) seconds later, its position vector relative to \(O\) is \(\left( t ^ { 2 } - 2 \right) \mathbf { i } + 2 t \mathbf { j }\), where the vectors \(\mathbf { i }\) and \(\mathbf { j }\) both have magnitude 1 metre and are directed parallel and perpendicular to the shoreline through \(O\).
    1. Find the speed with which the boat leaves \(O\).
    2. Show that the boat has constant acceleration and state the magnitude of this acceleration.
    3. Find the value of \(t\) when the boat is 40 m from \(O\).
    4. Comment on the limitations of the given model of the boat's motion.
    \includegraphics[max width=\textwidth, alt={}]{996976f3-2a97-4c68-8c97-f15a3bfde9a2-1_446_595_1965_349}
    The diagram shows a body which may be modelled as a uniform lamina. The body is suspended from the point marked \(A\) and rests in equilibrium.
    1. Calculate, to the nearest degree, the angle which the edge \(A B\) then makes with the vertical.
      (8 marks) Frank suggests that the angle between \(A B\) and the vertical would be smaller if the lamina were made from lighter material.
    2. State, with a brief explanation, whether Frank is correct.
      (2 marks) \section*{MECHANICS 2 (A) TEST PAPER 1 Page 2}
Edexcel M2 Q7
11 marks Moderate -0.3
7. A particle \(P\) moves in a straight line so that its displacement \(s\) metres from a fixed point \(O\) at time \(t\) seconds is given by the formula \(s = t ^ { 3 } - 7 t ^ { 2 } + 8 t\).
  1. Find the values of \(t\) when the velocity of \(P\) equals zero, and briefly describe what is happening to \(P\) at these times.
  2. Find the distance travelled by \(P\) between the times \(t = 3\) and \(t = 5\).
  3. Find the value of \(t\) when the acceleration of \(P\) is \(- 2 \mathrm {~ms} ^ { - 2 }\). Briefly explain the significance of a negative acceleration at this time.
Edexcel M2 Q3
7 marks Moderate -0.8
3. A particle \(P\) moves in a horizontal plane such that, at time \(t\) seconds, its velocity is \(\mathbf { v } \mathrm { ms } ^ { - 1 }\), where \(\mathbf { v } = 2 t \mathbf { i } - t ^ { \frac { 1 } { 2 } } \mathbf { j }\). When \(t = 0 , P\) is at the point with position vector \(- 10 \mathbf { i } + \mathbf { j }\) relative to a fixed origin \(O\).
  1. Find the position vector \(\mathbf { r }\) of \(P\) at time \(t\) seconds.
  2. Find the distance \(O P\) when \(t = 4\).
Edexcel M2 Q1
4 marks Moderate -0.8
  1. Particles of mass \(2 m , 3 m\) and \(5 m\) are placed at the points in the \(x - y\) plane with coordinates \(( - 1,5 ) , ( 0,6 )\) and \(( 3 , - 2 )\) respectively.
    Find the coordinates of the centre of mass of this system of particles.
  2. A lorry of mass 3800 kg is pulling a trailer of mass 1200 kg along a straight horizontal road. At a particular moment, the lorry and trailer are moving at a speed of \(10 \mathrm {~ms} ^ { - 1 }\) and accelerating at \(0.8 \mathrm {~ms} ^ { - 2 }\). The resistances to the motion of the lorry and the trailer are constant and of magnitude 1600 N and 600 N respectively.
    Find the rate, in kW , at which the engine of the lorry is working.
  3. A bullet of mass 0.05 kg is fired with speed \(u \mathrm {~ms} ^ { - 1 }\) from a gun, which recoils at a speed of \(0.008 u \mathrm {~ms} ^ { - 1 }\) in the opposite direction to that in which the bullet is fired.
    1. Find the mass of the gun.
    2. Find, in terms of \(u\), the kinetic energy given to the bullet and to the gun at the instant of firing.
    3. If the total kinetic energy created in firing the gun is 5100 J , find the value of \(u\).
    4. The acceleration of a particle \(P\) at time \(t \mathrm {~s}\) is \(\mathbf { a } \mathrm { ms } ^ { - 2 }\), where \(\mathbf { a } = 4 \mathrm { e } ^ { t } \mathbf { i } - \mathrm { e } ^ { t } \mathbf { j }\). When \(t = 0 , P\) has velocity \(4 \mathrm { i } \mathrm { ms } ^ { - 1 }\).
    1. Find the speed of \(P\) when \(t = 2\).
    2. Find the time at which the direction of motion of \(P\) is parallel to the vector \(5 \mathbf { i } - \mathbf { j }\).
    \includegraphics[max width=\textwidth, alt={}]{63133ab4-9381-4777-a575-1207219948b7-1_323_383_1992_429}
    A uniform plank \(A B\), of mass 3 kg and length 2 m , rests in equilibrium with the point \(P\) in contact with a smooth cylinder. The end \(B\) rests on a rough horizontal surface and the coefficient of friction between the plank and the surface is \(\frac { 1 } { 3 } . A B\) makes an angle of \(60 ^ { \circ }\) with the horizontal.
    If the plank is in limiting equilibrium in this position, find
    1. the magnitude of the force exerted by the cylinder on the plank at \(P\),
    2. the distance \(A P\). \section*{MECHANICS 2 (A) TEST PAPER 10 Page 2}
Edexcel M2 Q6
11 marks Moderate -0.3
  1. Two smooth spheres \(A\) and \(B\) have equal radii and masses 0.4 kg and 0.8 kg respectively. They are moving in opposite directions along the same straight line, with speeds \(3 \mathrm {~ms} ^ { - 1 }\) and 2 \(\mathrm { ms } ^ { - 1 }\) respectively, and collide directly. The coefficient of restitution between \(A\) and \(B\) is 0.8 .
    1. Calculate the speeds of \(A\) and \(B\) after the impact, stating in each case whether the direction of motion has been reversed.
    2. Find the kinetic energy, in J, lost in the impact.
    3. A point of light, \(P\), is moving along a straight line in such a way that, \(t\) seconds after passing through a fixed point \(O\) on the line, its velocity is \(v \mathrm {~ms} ^ { - 1 }\), where \(v = \frac { 1 } { 2 } t ^ { 2 } - 4 t + 10\). Calculate
    1. the velocity of \(P 6\) seconds after it passes \(O\),
    2. the magnitude of the acceleration of \(P\) when \(t = 1\),
    3. the minimum speed of \(P\),
    4. the distance travelled by \(P\) in the fourth second after it passes \(O\).
    5. A bullet is fired out of a window at a height of 5.2 m above horizontal ground. The initial velocity of the bullet is \(392 \mathrm {~ms} ^ { - 1 }\) at an angle \(\alpha\) above the vertical, where \(\sin \alpha = \frac { 1 } { 20 }\), as shown.
      Find \includegraphics[max width=\textwidth, alt={}, center]{63133ab4-9381-4777-a575-1207219948b7-2_335_490_1343_1419}
    1. the range of times after firing during which the bullet is 15 m or more above ground level,
    2. the greatest height above the ground reached by the bullet,
    3. the horizontal distance travelled by the bullet before it reaches its highest point.
    Certain modelling assumptions have been made about the bullet.
  2. State these assumptions and suggest a way in which the model could be refined.
  3. State, with a reason, whether you think this refinement would make a significant difference to the answers.
    (2 marks)
Edexcel M2 Q4
11 marks Standard +0.3
4. A particle \(P\) moves in a straight horizontal line such that its acceleration at time \(t\) seconds is proportional to \(\left( 3 t ^ { 2 } - 5 \right)\). Given that at time \(t = 0 , P\) is at rest at the origin \(O\) and that at time \(t = 3\), its velocity is \(3 \mathrm {~ms} ^ { - 1 }\),
  1. find, in \(\mathrm { m } \mathrm { s } ^ { - 2 }\), the acceleration of \(P\) in terms of \(t\),
  2. show that the displacement of the particle, \(s\) metres, from \(O\) at time \(t\) is given by $$s = \frac { 1 } { 16 } t ^ { 2 } \left( t ^ { 2 } - 10 \right)$$ (4 marks)
Edexcel M2 Q3
9 marks Moderate -0.3
3. A particle moves in a straight horizontal line such that its velocity, \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), at time \(t\) seconds is given by \(v = 2 t ^ { 2 } - 9 t + 4\). Initially, the particle has displacement 9 m from a fixed point \(O\) on the line.
  1. Find the initial velocity of the particle.
  2. Show that the particle is at rest when \(t = 4\) and find the other value of \(t\) when it is at rest.
  3. Find the displacement of the particle from \(O\) when \(t = 6\).
Edexcel M2 Q5
13 marks Standard +0.3
5. A particle \(P\) moves in a straight line with an acceleration of \(( 6 t - 10 ) \mathrm { m } \mathrm { s } ^ { - 2 }\) at time \(t\) seconds. Initially \(P\) is at \(O\), a fixed point on the line, and has velocity \(3 \mathrm {~ms} ^ { - 1 }\).
  1. Find the values of \(t\) for which the velocity of \(P\) is zero.
  2. Show that, during the first two seconds, \(P\) travels a distance of \(6 \frac { 26 } { 27 } \mathrm {~m}\).
Edexcel M2 Q2
6 marks Moderate -0.3
2. A particle \(P\) moves along the \(x\)-axis such that its displacement, \(x\) metres, from the origin \(O\) at time \(t\) seconds is given by $$x = 2 + t - \frac { 1 } { 10 } \mathrm { e } ^ { t }$$
  1. Find the distance of \(P\) from \(O\) when \(t = 0\).
  2. Find, correct to 1 decimal place, the value of \(t\) when the velocity of \(P\) is zero.
    (4 marks)
Edexcel M2 Q3
8 marks Moderate -0.8
3. A particle moves along a straight horizontal track such that its displacement, \(s\) metres, from a fixed point \(O\) on the line after \(t\) seconds is given by $$s = 2 t ^ { 3 } - 13 t ^ { 2 } + 20 t$$
  1. Find the values of \(t\) for which the particle is at \(O\).
  2. Find the values of \(t\) at which the particle comes instantaneously to rest.
OCR M3 2012 June Q3
10 marks Standard +0.3
3 A particle \(P\) of mass \(m \mathrm {~kg}\) is released from rest and falls vertically. When \(P\) has fallen a distance of \(x \mathrm {~m}\) it has a speed of \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The only forces acting on \(P\) are its weight and air resistance of magnitude \(\frac { 1 } { 400 } m v ^ { 2 } \mathrm {~N}\).
  1. Find \(v ^ { 2 }\) in terms of \(x\) and show that \(v ^ { 2 }\) must be less than 3920 .
  2. Find the speed of \(P\) when it has fallen 100 m .
OCR M3 2013 June Q3
11 marks Standard +0.3
3 A particle \(P\) of mass 0.2 kg moves on a smooth horizontal plane. Initially it is projected with velocity \(0.8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from a fixed point \(O\) towards another fixed point \(A\). At time \(t\) s after projection, \(P\) is \(x \mathrm {~m}\) from \(O\) and is moving with velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), with the direction \(O A\) being positive. A force of \(( 1.5 t - 1 ) \mathrm { N }\) acts on \(P\) in the direction parallel to \(O A\).
  1. Find an expression for \(v\) in terms of \(t\).
  2. Find the time when the velocity of \(P\) is next \(0.8 \mathrm {~ms} ^ { - 1 }\).
  3. Find the times when \(P\) subsequently passes through \(O\).
  4. Find the distance \(P\) travels in the third second of its motion.