Acceleration from velocity differentiation

7 A particle \(P\) travels in a straight line from \(A\) to \(D\), passing through the points \(B\) and \(C\). For the section \(A B\) the velocity of the particle is \(\left( 0.5 t - 0.01 t ^ { 2 } \right) \mathrm { m } \mathrm { s } ^ { - 1 }\), where \(t \mathrm {~s}\) is the time after leaving \(A\).

1. Given that the acceleration of \(P\) at \(B\) is \(0.1 \mathrm {~m} \mathrm {~s} ^ { - 2 }\), find the time taken for \(P\) to travel from \(A\) to \(B\). The acceleration of \(P\) from \(B\) to \(C\) is constant and equal to \(0.1 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
2. Given that \(P\) reaches \(C\) with speed \(14 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), find the time taken for \(P\) to travel from \(B\) to \(C\).
   \(P\) travels with constant deceleration \(0.3 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) from \(C\) to \(D\). Given that the distance \(C D\) is 300 m , find
3. the speed with which \(P\) reaches \(D\),
4. the distance \(A D\). \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. University of 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. }

2. A particle \(P\) moves in a straight line. At time \(t = 0 , P\) passes through a point \(O\) on the line. At time \(t\) seconds, the velocity of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) where

1. Find the acceleration of \(P\) when \(t = \frac { 1 } { 2 }\)
2. Find the distance travelled by \(P\) in the interval \(0 \leqslant t \leqslant 1\) At time \(t\) seconds, the velocity of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) where $$v = ( 2 t - 1 ) ( 1 - t )$$
3. Find the acceleration of \(P\) when \(t = \frac { 1 } { 2 }\)

3 Fig. 3 is a sketch of the velocity-time graph modelling the velocity of a sprinter at the start of a race. \begin{figure}[h] \end{figure}

1. How can you tell from the sketch that the acceleration is not modelled as being constant for \(0 \leqslant t \leqslant 4\) ? The velocity of the sprinter, \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), for the time interval \(0 \leqslant t \leqslant 4\) is modelled by the expression $$v = 3 t - \frac { 3 } { 8 } t ^ { 2 }$$
2. Find the acceleration that the model predicts for \(t = 4\) and comment on what this suggests about the running of the sprinter.
3. Calculate the distance run by the sprinter from \(t = 1\) to \(t = 4\).

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 moves along a straight line. Its velocity \(v \mathrm {~ms} ^ { - 1 }\) at time \(t\) s is given by \(\mathbf { v } = 2 \mathbf { t } + 0.6 \mathbf { t } ^ { 2 }\).
Find an expression for the acceleration of the particle at time \(t\).

5 Fig. 3 is a sketch of the velocity-time graph modelling the velocity of a sprinter at the start of a race. \begin{figure}[h] \end{figure}

1. How can you tell from the sketch that the acceleration is not modelled as being constant for \(0 \leqslant t \leqslant 4\) ? The velocity of the sprinter, \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), for the time interval \(0 \leqslant t \leqslant 4\) is modelled by the expression $$v = 3 t - \frac { 3 } { 8 } t ^ { 2 } .$$
2. Find the acceleration that the model predicts for \(t = 4\) and comment on what this suggests about the running of the sprinter.
3. Calculate the distance run by the sprinter from \(t = 1\) to \(t = 4\).

5 Fig. 3 is a sketch of the velocity-time graph modelling the velocity of a sprinter at the start of a race. \begin{figure}[h] \end{figure}

1. How can you tell from the sketch that the acceleration is not modelled as being constant for \(0 \leqslant t \leqslant 4\) ? The velocity of the sprinter, \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), for the time interval \(0 \leqslant t \leqslant 4\) is modelled by the expression $$v = 3 t - \frac { 3 } { 8 } t ^ { 2 } .$$
2. Find the acceleration that the model predicts for \(t = 4\) and comment on what this suggests about the running of the sprinter.
3. Calculate the distance run by the sprinter from \(t = 1\) to \(t = 4\).

3 A particle moves in a straight line and at time \(t\) has velocity \(v\), where $$v = 2 t - 12 \mathrm { e } ^ { - t } , \quad t \geqslant 0$$

1. 1. Find an expression for the acceleration of the particle at time \(t\).
   2. State the range of values of the acceleration of the particle.
2. When \(t = 0\), the particle is at the origin. Find an expression for the displacement of the particle from the origin at time \(t\).

2 A particle moves in a straight line and at time \(t\) it has velocity \(v\), where $$v = 3 t ^ { 2 } - 2 \sin 3 t + 6$$

1. 1. Find an expression for the acceleration of the particle at time \(t\).
   2. When \(t = \frac { \pi } { 3 }\), show that the acceleration of the particle is \(2 \pi + 6\).
2. When \(t = 0\), the particle is at the origin. Find an expression for the displacement of the particle from the origin at time \(t\).

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. 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 \(t\).

2. A particle \(P\) of mass 0.25 kg is moving on a horizontal plane. At time \(t\) seconds the velocity, \(\mathbf { v } \mathrm { ms } ^ { - 1 }\), of \(P\) relative to a fixed origin \(O\) is given by $$\mathbf { v } = \ln ( t + 1 ) \mathbf { i } - \mathrm { e } ^ { - 2 t } \mathbf { j } , t \leq 0 ,$$ where \(\mathbf { i }\) and \(\mathbf { j }\) are perpendicular unit vectors in the horizontal plane.

1. Find the acceleration of \(P\) in terms of \(t\).
2. Find, correct to 3 significant figures, the magnitude of the resultant force acting on \(P\) when \(t = 1\).
   (4 marks)

1. The velocity, \(\mathbf { v ~ c m ~ s } { } ^ { - 1 }\), at time \(t\) seconds, of a radio-controlled toy is modelled by the formula

$$\mathbf { v } = \mathrm { e } ^ { 2 t } \mathbf { i } + 2 t \mathbf { j } ,$$ where \(\mathbf { i }\) and \(\mathbf { j }\) are perpendicular unit vectors.

1. Find the acceleration of the toy in terms of \(t\).
2. Find, correct to 2 significant figures, the time at which the acceleration of the toy is parallel to the vector \(( 4 \mathbf { i } + \mathbf { j } )\).
3. Explain why this model is unlikely to be realistic for large values of \(t\).

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

$$v = \frac { 1 } { 2 } \left( 3 \mathrm { e } ^ { 2 t } - 1 \right) \quad t \geqslant 0$$ The acceleration of \(P\) at time \(t\) seconds is \(a \mathrm {~ms} ^ { - 2 }\)

1. Show that \(a = 2 v + 1\)
2. Find the acceleration of \(P\) when \(t = 0\)
3. Find the exact distance travelled by \(P\) in accelerating from a speed of \(1 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) to a speed of \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\)

3 A particle moves in a straight line and at time \(t\) has velocity \(v\), where $$v = 2 t - 12 \mathrm { e } ^ { - t } , \quad t \geqslant 0$$

1. 1. Find an expression for the acceleration of the particle at time \(t\).
   2. State the range of values of the acceleration of the particle.
2. When \(t = 0\), the particle is at the origin. Find an expression for the displacement of the particle from the origin at time \(t\).
   (4 marks)

1 A particle moves in a straight line and at time \(t\) seconds has velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), where $$v = 6 t ^ { 2 } + 4 t - 7 , \quad t \geqslant 0$$

1. Find an expression for the acceleration of the particle at time \(t\).
2. The mass of the particle is 3 kg . Find the resultant force on the particle when \(t = 4\).
3. When \(t = 0\), the displacement of the particle from the origin is 5 metres. Find an expression for the displacement of the particle from the origin at time \(t\).

16 A toy remote control speed boat is launched from one edge of a small pond and moves in a straight line across the pond's surface. The boat's velocity, \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), is modelled in terms of time, \(t\) seconds after the boat is launched, by the expression $$v = 0.9 + 0.16 t - 0.06 t ^ { 2 }$$ 16

1. Find the acceleration of the boat when \(t = 2\)
   16
2. Find the displacement of the boat, from the point where it was launched, when \(t = 2\)