Non-constant acceleration

4 A particle \(P\) moves in a straight line. At time \(t\) seconds after starting from rest at the point \(O\) on the line, the acceleration of \(P\) is \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\), where \(a = 0.075 t ^ { 2 } - 1.5 t + 5\).

1. Find an expression for the displacement of \(P\) from \(O\) in terms of \(t\).
2. Hence find the time taken for \(P\) to return to the point \(O\).

6 In this question the origin is a point on the ground. The directions of the unit vectors \(\left( \begin{array} { l } 1 \\ 0 \\ 0 \end{array} \right) , \left( \begin{array} { l } 0 \\ 1 \\ 0 \end{array} \right)\) and \(\left( \begin{array} { l } 0 \\ 0 \\ 1 \end{array} \right)\) are
\includegraphics[max width=\textwidth, alt={}, center]{63a2dc41-5e8b-4275-8653-ece5067c4306-5_398_689_434_689} Alesha does a sky-dive on a day when there is no wind. The dive starts when she steps out of a moving helicopter. The dive ends when she lands gently on the ground.

• During the dive Alesha can reduce the magnitude of her acceleration in the vertical direction by spreading her arms and increasing air resistance.
• During the dive she can use a power unit strapped to her back to give herself an acceleration in a horizontal direction.
• Alesha's mass, including her equipment, is 100 kg .
• Initially, her position vector is \(\left( \begin{array} { r } - 75 \\ 90 \\ 750 \end{array} \right) \mathrm { m }\) and her velocity is \(\left( \begin{array} { r } - 5 \\ 0 \\ - 10 \end{array} \right) \mathrm { ms } ^ { - 1 }\).
  1. Calculate Alesha's initial speed, and the initial angle between her motion and the downward vertical.

At a certain time during the dive, forces of \(\left( \begin{array} { r } 0 \\ 0 \\ - 980 \end{array} \right) \mathrm { N } , \left( \begin{array} { r } 0 \\ 0 \\ 880 \end{array} \right) \mathrm { N }\) and \(\left( \begin{array} { r } 50 \\ - 20 \\ 0 \end{array} \right) \mathrm { N }\) are acting on Alesha.

Suggest how these forces could arise.

Find Alesha's acceleration at this time, giving your answer in vector form, and show that, correct to 3 significant figures, its magnitude is \(1.14 \mathrm {~ms} ^ { - 2 }\). One suggested model for Alesha's motion is that the forces on her are constant throughout the dive from when she leaves the helicopter until she reaches the ground.

Find expressions for her velocity and position vector at time \(t\) seconds after the start of the dive according to this model. Verify that when \(t = 30\) she is at the origin.

Explain why consideration of Alesha's landing velocity shows this model to be unrealistic.

1. At time \(t = 0\), a toy electric car is at rest at a fixed point \(O\). The car then moves in a horizontal straight line so that at time \(t\) seconds \(( t > 0 )\) after leaving \(O\), the velocity of the car is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the acceleration of the car is modelled as \(( p + q v ) \mathrm { ms } ^ { - 2 }\), where \(p\) and \(q\) are constants.

When \(t = 0\), the acceleration of the car is \(3 \mathrm {~ms} ^ { - 2 }\)
When \(t = T\), the acceleration of the car is \(\frac { 1 } { 2 } \mathrm {~ms} ^ { - 2 }\) and \(v = 4\)

1. Show that $$8 \frac { \mathrm {~d} v } { \mathrm {~d} t } = ( 24 - 5 v )$$
2. Find the exact value of \(T\), simplifying your answer.

3 The area of a circular stain is growing at a rate of \(1 \mathrm {~mm} ^ { 2 }\) per second. Find the rate of increase of its radius at an instant when its radius is 2 mm .

4 When the gas in a balloon is kept at a constant temperature, the pressure \(P\) in atmospheres and the volume \(V \mathrm {~m} ^ { 3 }\) are related by the equation $$P = \frac { k } { V }$$ where \(k\) is a constant. [This is known as Boyle's Law.]
When the volume is \(100 \mathrm {~m} ^ { 3 }\), the pressure is 5 atmospheres, and the volume is increasing at a rate of \(10 \mathrm {~m} ^ { 3 }\) per second.

1. Show that \(k = 500\).
2. Find \(\frac { \mathrm { d } P } { \mathrm {~d} V }\) in terms of \(V\).
3. Find the rate at which the pressure is decreasing when \(V = 100\).

7 The point \(P ( x , y )\) is moving along the curve \(y = x ^ { 2 } - \frac { 10 } { 3 } x ^ { \frac { 3 } { 2 } } + 5 x\) in such a way that the rate of change of \(y\) is constant. Find the values of \(x\) at the points at which the rate of change of \(x\) is equal to half the rate of change of \(y\).

2 A particle moves along a straight line containing a point O . Its displacement, \(x \mathrm {~m}\), from O at time \(t\) seconds is given by $$x = 12 t - t ^ { 3 } , \text { where } - 10 \leqslant t \leqslant 10$$ Find the values of \(x\) for which the velocity of the particle is zero.

3 The velocity-time graph for the motion of a particle is shown below. The velocity \(v \mathrm {~ms} ^ { - 1 }\) at time \(t \mathrm {~s}\) is given by \(\mathrm { v } = - \mathrm { t } ^ { 2 } + 6 \mathrm { t } - 6\) where \(0 \leqslant t \leqslant 5\).
\includegraphics[max width=\textwidth, alt={}, center]{7af62e61-c67f-4d05-b6b9-c1a110345812-3_860_979_1082_239}

1. Find the times at which the velocity is \(2 \mathrm {~ms} ^ { - 1 }\).
2. Write down the greatest speed of the particle.

2 A particle \(P\) moves in a straight line. At time \(t\) s after leaving a point \(O\) on the line, \(P\) has velocity \(v \mathrm {~ms} ^ { - 1 }\), where \(\mathrm { v } = 44 \mathrm { t } - 6 \mathrm { t } ^ { 2 } - 36\).

1. Find the set of values of \(t\) for which the acceleration of the particle is positive.
2. Find the two values of \(t\) at which \(P\) returns to \(O\).
   \includegraphics[max width=\textwidth, alt={}, center]{3eaf3652-ff91-4bae-9f20-83487d635612-04_714_796_248_635} Four coplanar forces of magnitude \(P \mathrm {~N} , 10 \mathrm {~N} , 16 \mathrm {~N}\) and 2 N act at a point in the directions shown in the diagram. It is given that the forces are in equilibrium. Find the values of \(\theta\) and \(P\).

2 A particle starts from rest at \(O\) and travels in a straight line. Its acceleration is \(( 3 - 2 x ) \mathrm { ms } ^ { - 2 }\), where \(x \mathrm {~m}\) is the displacement of the particle from \(O\).

1. Find the value of \(x\) for which the velocity of the particle reaches its maximum value.
2. Find this maximum velocity.

6 A car, of mass \(m\), is moving along a straight smooth horizontal road. At time \(t\), the car has speed \(v\). As the car moves, it experiences a resistance force of magnitude \(0.05 m v\). No other horizontal force acts on the car.

1. Show that $$\frac { \mathrm { d } v } { \mathrm {~d} t } = - 0.05 v$$
2. When \(t = 0\), the speed of the car is \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Show that \(v = 20 \mathrm { e } ^ { - 0.05 t }\).
3. Find the time taken for the speed of the car to reduce to \(10 \mathrm {~ms} ^ { - 1 }\).

6 A particle \(P\) of mass 2 kg moving on a horizontal straight line has displacement \(x \mathrm {~m}\) from a fixed point \(O\) on the line and velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at time \(t \mathrm {~s}\). The only horizontal force acting on \(P\) has magnitude \(\frac { 1 } { 10 } ( 2 \mathrm { v } - 1 ) ^ { 2 } \mathrm { e } ^ { - \mathrm { t } } \mathrm { N }\) and acts towards \(O\). When \(t = 0 , x = 1\) and \(v = 3\).

1. Find an expression for \(v\) in terms of \(t\).
   \includegraphics[max width=\textwidth, alt={}, center]{b57762bf-7a4f-486d-b9f2-8ae727bfb630-12_69_1569_466_328}
2. Find an expression for \(x\) in terms of \(t\).

1 A particle moves along a straight line. At time \(t\), it has velocity \(v\), where $$v = 4 t ^ { 3 } - 8 \sin 2 t + 5$$ When \(t = 0\), the particle is at the origin.
Find an expression for the displacement of the particle from the origin at time \(t\).

5 A particle \(P\) moves along the \(x\)-axis in the positive direction. The velocity of \(P\) at time \(t \mathrm {~s}\) is \(0.03 t ^ { 2 } \mathrm {~m} \mathrm {~s} ^ { - 1 }\). When \(t = 5\) the displacement of \(P\) from the origin \(O\) is 2.5 m .

1. Find an expression, in terms of \(t\), for the displacement of \(P\) from \(O\).
2. Find the velocity of \(P\) when its displacement from \(O\) is 11.25 m .

1. A particle \(P\) moves on the \(x\)-axis. At time \(t\) seconds, \(t \geqslant 0 , P\) is \(x\) metres from the origin \(O\) and moving with velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in the direction of \(x\) increasing, where

$$v = 5 \sin 2 t$$ When \(t = 0 , x = 1\) and \(P\) is at rest.

1. Find the magnitude and direction of the acceleration of \(P\) at the instant when \(P\) is next at rest.
2. Show that \(1 \leqslant x \leqslant 6\)
3. Find the total time, in the first \(4 \pi\) seconds of the motion, for which \(P\) is more than 3 metres from \(O\)
   \includegraphics[max width=\textwidth, alt={}]{a7901165-1679-4d30-9444-0c27020e32ea-16_2260_52_309_1982}

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)

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 \(( - \mathbf { i } - 3 \mathbf { j } )\).

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 .

5. At time \(t\) seconds ( \(t \geqslant 0\) ), a particle \(P\) has velocity \(\mathbf { v m ~ s } ^ { - 1 }\), where $$\mathbf { v } = \left( 3 t ^ { 2 } - 9 t + 6 \right) \mathbf { i } + \left( t ^ { 2 } + t - 6 \right) \mathbf { j }$$

1. Find the acceleration of \(P\) when \(t = 3\) When \(t = 0 , P\) is at the fixed point \(O\).
   The particle comes to instantaneous rest at the point \(A\).
2. Find the distance \(O A\).

5. A particle \(P\) moves in a horizontal plane. The acceleration of \(P\) is \(( - \mathbf { i } + 2 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 2 }\). At time \(t = 0\), the velocity of \(P\) is \(( 2 \mathbf { i } - 3 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\).

1. Find, to the nearest degree, the angle between the vector \(\mathbf { j }\) and the direction of motion of \(P\) when \(t = 0\). At time \(t\) seconds, the velocity of \(P\) is \(\mathbf { v } \mathrm { m } \mathrm { s } ^ { - 1 }\). Find
2. an expression for \(\mathbf { v }\) in terms of \(t\), in the form \(a \mathbf { i } + b \mathbf { j }\),
3. the speed of \(P\) when \(t = 3\),
4. the time when \(P\) is moving parallel to \(\mathbf { i }\).

6 In this question you must show detailed reasoning. In this question, positions are given relative to a fixed origin, O . The unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are in the \(x\) - and \(y\)-directions respectively in a horizontal plane. Distances are measured in centimetres and the time, \(t\), is measured in seconds, where \(0 \leqslant t \leqslant 5\). A small radio-controlled toy car C moves on a horizontal surface which contains O.
The acceleration of C is given by \(2 \mathbf { i } + t \mathbf { j } \mathrm {~cm} \mathrm {~s} ^ { - 2 }\).
When \(t = 4\), the displacement of C from O is \(16 \mathbf { i } + \frac { 32 } { 3 } \mathbf { j } \mathrm {~cm}\), and the velocity of C is \(8 \mathbf { i } \mathrm {~cm} \mathrm {~s} ^ { - 1 }\).
Determine a cartesian equation for the path of C for \(0 \leqslant t \leqslant 5\). You are not required to simplify your answer.

2. A particle \(P\) of mass 0.5 kg moves under the action of a single force \(\mathbf { F }\) newtons. At time \(t\) seconds, the velocity \(\mathbf { v } \mathrm { m } \mathrm { s } ^ { - 1 }\) of \(P\) is given by $$\mathbf { v } = 3 t ^ { 2 } \mathbf { i } + ( 1 - 4 t ) \mathbf { j }$$ Find

1. the acceleration of \(P\) at time \(t\) seconds,
2. the magnitude of \(\mathbf { F }\) when \(t = 2\).

2. A particle \(P\) of mass 1.5 kg is moving under the action of a constant force ( \(3 \mathbf { i } - 7.5 \mathbf { j }\) ) N. Initially \(P\) has velocity \(( 2 \mathbf { i } + 3 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\). Find

1. the magnitude of the acceleration of \(P\),
2. the velocity of \(P\), in terms of \(\mathbf { i }\) and \(\mathbf { j }\), when \(P\) has been moving for 4 seconds.

12 In this question the unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are in the directions east and north respectively.
A particle \(P\) is moving on a smooth horizontal surface under the action of a single force \(\mathbf { F N }\). At time \(t\) seconds, where \(t \geqslant 0\), the velocity \(\mathbf { v } \mathrm { ms } ^ { - 1 }\) of \(P\), relative to a fixed origin \(O\), is given by \(\mathbf { v } = ( 1 - 2 t ) \mathbf { i } + \left( 2 t ^ { 2 } + t - 13 \right) \mathbf { j }\).

1. Show that \(P\) is never stationary.
2. Find, in terms of \(\mathbf { i }\) and \(\mathbf { j }\), the acceleration of \(P\) at time \(t\). The mass of \(P\) is 0.5 kg .
3. Determine the magnitude of \(\mathbf { F }\) when \(P\) is moving in the direction of the vector \(- 2 \mathbf { i } + \mathbf { j }\). Give your answer correct to \(\mathbf { 3 }\) significant figures. When \(t = 1 , P\) is at the point with position vector \(\frac { 1 } { 6 } \mathbf { j }\).
4. Determine the bearing of \(P\) from \(O\) at time \(t = 1.5\).

3 \(4 t \quad t t t v t \quad t\)
\(t v\) tt \(t t\) tttvt \(t \quad t \quad t \quad t t \quad t t\)

1. \(x t t \quad t t \quad\) t \(t\) - • • •
2. \(t v\) t \(t\)
3. \(t v\) t \(t v\)
   \(5 t v t v v t\)
4. \(t\) t \(t\)
5. \(t t\)

4 A cyclist travels along a straight road. Her velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), at time \(t\) seconds after starting from a point \(O\), is given by $$\begin{aligned} & v = 2 \quad \text { for } 0 \leqslant t \leqslant 10 \\ & v = 0.03 t ^ { 2 } - 0.3 t + 2 \quad \text { for } t \geqslant 10 . \end{aligned}$$

1. Find the displacement of the cyclist from \(O\) when \(t = 10\).
2. Show that, for \(t \geqslant 10\), the displacement of the cyclist from \(O\) is given by the expression \(0.01 t ^ { 3 } - 0.15 t ^ { 2 } + 2 t + 5\).
3. Find the time when the acceleration of the cyclist is \(0.6 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). Hence find the displacement of the cyclist from \(O\) when her acceleration is \(0.6 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).

7 A particle moves in a straight line starting from a point \(O\) before coming to instantaneous rest at a point \(X\). At time \(t \mathrm {~s}\) after leaving \(O\), the velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) of the particle is given by $$\begin{array} { l l } v = 7.2 t ^ { 2 } & 0 \leqslant t \leqslant 2 , \\ v = 30.6 - 0.9 t & 2 \leqslant t \leqslant 8 , \\ v = \frac { 1600 } { t ^ { 2 } } + k t & 8 \leqslant t , \end{array}$$ where \(k\) is a constant. It is given that there is no instantaneous change in velocity at \(t = 8\).
Find the distance \(O X\).
If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

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 \(t = 8\).

5 In a science fiction story a new type of spaceship travels to the moon. The journey takes place along a straight line. The spaceship starts from rest on the earth and arrives at the moon's surface with zero speed. Its speed, \(v\) kilometres per hour at time \(t\) hours after it has started, is given by $$v = 37500 \left( 4 t - t ^ { 2 } \right) .$$

1. Show that the spaceship takes 4 hours to reach the moon.
2. Find an expression for the distance the spaceship has travelled at time \(t\). Hence find the distance to the moon.
3. Find the spaceship's greatest speed during the journey. Section B (36 marks)

5. A particle moves along the \(x\)-axis. At time \(t\) seconds, \(t \geqslant 0\), the velocity of the particle is \(v \mathrm {~ms} ^ { - 1 }\) in the direction of \(x\) increasing, where \(v = 2 t ^ { \frac { 3 } { 2 } } - 6 t + 2\) At time \(t = 0\) the particle passes through the origin \(O\). At the instant when the acceleration of the particle is zero, the particle is at the point \(A\). Find the distance \(O A\).
(8)

14 A particle, P , is moving along a straight line such that its acceleration \(a \mathrm {~ms} ^ { - 2 }\), at any time, \(t\) seconds, may be modelled by $$a = 3 + 0.2 t$$ When \(t = 2\), the velocity of P is \(k \mathrm {~m} \mathrm {~s} ^ { - 1 }\)
14

1. Show that the initial velocity of P is given by the expression \(( k - 6.4 ) \mathrm { ms } ^ { - 1 }\)
   [0pt] [4 marks]
   14
2. The initial velocity of P is one fifth of the velocity when \(t = 2\) Find the value of \(k\).

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
（a）the value of \(c\) ， （b）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 \(t = 1.5\) ．

12
\includegraphics[max width=\textwidth, alt={}, center]{31b0d5b6-1593-489b-bbcd-486e7c96ff18-09_647_935_260_242} A particle \(P\) moves in a straight line. At time \(t\) seconds, where \(t \geqslant 0\), the velocity of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). It is given that \(v = - 3 t ^ { 2 } + 24 t + k\), where \(k\) is a positive constant. The diagram shows the velocity-time graph for the motion of \(P\).
\(P\) attains its maximum velocity at time \(T\) seconds. Given that the distance travelled by \(P\) between times \(t = 1\) and \(t = T\) is 297 m , determine the time when \(P\) is instantaneously at rest. \section*{END OF QUESTION PAPER}

2 A particle \(P\) moves with acceleration \(( - 3 \mathbf { i } + 12 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 2 }\). Initially the velocity of \(P\) is \(4 \mathbf { i } \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Find the velocity of \(P\) at time \(t\) seconds.
2. Find the speed of \(P\) when \(t = 0.5\).

1 A particle travels along a straight line. Its acceleration during the time interval \(0 \leqslant t \leqslant 8\) is given by the acceleration-time graph in Fig. 1. \begin{figure}[h] \end{figure}

1. Write down the acceleration of the particle when \(t = 4\). Given that the particle starts from rest, find its speed when \(t = 4\).
2. Write down an expression in terms of \(t\) for the acceleration, \(a \mathrm {~ms} ^ { - 2 }\), of the particle in the time interval \(0 \leqslant t \leqslant 4\).
3. Without calculation, state the time at which the speed of the particle is greatest. Give a reason for your answer.
4. Calculate the change in speed of the particle from \(t = 5\) to \(t = 8\), indicating whether this is an increase or a decrease.

8 A particle of mass 4 kg moves horizontally in a straight line. At time \(t\) seconds the velocity of the particle is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\)
The following horizontal forces act on the particle:

• a constant driving force of magnitude 1.8 newtons
• another driving force of magnitude \(30 \sqrt { t }\) newtons
• a resistive force of magnitude \(0.08 v ^ { 2 }\) newtons

When \(t = 70 , v = 54\)
Use Euler's method with a step length of 0.5 to estimate the velocity of the particle after 71 seconds. Give your answer to four significant figures.

1 A particle moves in a straight line. The displacement of the particle at time \(t \mathrm {~s}\) is \(s \mathrm {~m}\), where $$s = t ^ { 3 } - 6 t ^ { 2 } + 4 t$$ Find the velocity of the particle at the instant when its acceleration is zero.

1 A particle moves along a straight line through the origin. At time \(t\), the displacement, \(s\), of the particle from the origin is given by $$s = 5 t ^ { 2 } + 3 \cos 4 t$$ Find the velocity of the particle at time \(t\).

17 A particle is moving such that its position vector, \(\mathbf { r }\) metres, at time \(t\) seconds, is given by $$\mathbf { r } = \mathrm { e } ^ { t } \cos t \mathbf { i } + \mathrm { e } ^ { t } \sin t \mathbf { j }$$ Show that the magnitude of the acceleration of the particle, \(a \mathrm {~ms} ^ { - 2 }\), is given by $$a = 2 \mathrm { e } ^ { t }$$ Fully justify your answer.
\includegraphics[max width=\textwidth, alt={}, center]{ad6590e8-6673-45ca-bef3-a14716978827-27_2490_1728_217_141}

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 \(t\).

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 \(t\).

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.

6 Two particles \(A\) and \(B\) start to move at the same instant from a point \(O\). The particles move in the same direction along the same straight line. The acceleration of \(A\) at time \(t \mathrm {~s}\) after starting to move is \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\), where \(a = 0.05 - 0.0002 t\).

1. Find A's velocity when \(t = 200\) and when \(t = 500\).
   \(B\) moves with constant acceleration for the first 200 s and has the same velocity as \(A\) when \(t = 200 . B\) moves with constant retardation from \(t = 200\) to \(t = 500\) and has the same velocity as \(A\) when \(t = 500\).
2. Find the distance between \(A\) and \(B\) when \(t = 500\).

4. At time \(t\) seconds the velocity of a particle \(P\) is \([ ( 4 t - 5 ) \mathbf { i } + 3 \mathbf { j } ] \mathrm { m } \mathrm { s } ^ { - 1 }\). When \(t = 0\), the position vector of \(P\) is \(( 2 \mathbf { i } + 5 \mathbf { j } ) \mathrm { m }\), relative to a fixed origin \(O\).

1. Find the value of \(t\) when the velocity of \(P\) is parallel to the vector \(\mathbf { j }\).
2. Find an expression for the position vector of \(P\) at time \(t\) seconds. A second particle \(Q\) moves with constant velocity \(( - 2 \mathbf { i } + c \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\). When \(t = 0\), the position vector of \(Q\) is \(( 11 \mathbf { i } + 2 \mathbf { j } ) \mathrm { m }\). The particles \(P\) and \(Q\) collide at the point with position vector ( \(d \mathbf { i } + 14 \mathbf { j }\) ) m.
3. Find
   1. the value of \(c\),
   2. the value of \(d\).

5 Particles \(X\) and \(Y\) move in a straight line through points \(A\) and \(B\). Particle \(X\) starts from rest at \(A\) and moves towards \(B\). At the same instant, \(Y\) starts from rest at \(B\). At time \(t\) seconds after the particles start moving

• the acceleration of \(X\) in the direction \(A B\) is given by \(( 12 t + 12 ) \mathrm { m } \mathrm { s } ^ { - 2 }\),
• the acceleration of \(Y\) in the direction \(A B\) is given by \(( 24 t - 8 ) \mathrm { m } \mathrm { s } ^ { - 2 }\).
  1. It is given that the velocities of \(X\) and \(Y\) are equal when they collide.

Calculate the distance \(A B\).

It is given instead that \(A B = 36 \mathrm {~m}\). Verify that \(X\) and \(Y\) collide after 3 s.

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 \(t = 2\).

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 \(t = 2\).

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.

3. A particle \(P\) moves along a straight line such that at time \(t\) seconds its velocity \(v \mathrm {~ms} ^ { - 1 }\) is given by: $$v ( t ) = t ^ { 2 } - 5 t + 4$$ Find the distance travelled by the particle between \(t = 1\) and \(t = 5.5\).
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2 A particle \(P\) moves in a straight line, starting from a point \(O\). At time \(t \mathrm {~s}\) after leaving \(O\), the velocity of \(P , v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), is given by \(v = 4 t ^ { 2 } - 8 t + 3\).

1. Find the two values of \(t\) at which \(P\) is at instantaneous rest.
2. Find the distance travelled by \(P\) between these two times.

8 A particle \(P\) moves in a straight line, passing through a point \(O\) with velocity \(4.2 \mathrm {~ms} ^ { - 1 }\). At time \(t \mathrm {~s}\) after \(P\) passes \(O\), the acceleration, \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\), of \(P\) is given by \(a = 0.6 t - 2.7\). Find the distance \(P\) travels between the times at which it is at instantaneous rest.
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5 The position vector \(\mathbf { r }\) metres of a particle at time \(t\) seconds is given by $$\mathbf { r } = \left( 1 + 12 t - 2 t ^ { 2 } \right) \mathbf { i } + \left( t ^ { 2 } - 6 t \right) \mathbf { j }$$

1. Find an expression for the velocity of the particle at time \(t\).
2. Determine whether the particle is ever stationary.

1
\includegraphics[max width=\textwidth, alt={}, center]{155bc571-80e4-4c93-859f-bb150a109211-2_675_1380_255_379} A woman walks in a straight line. The woman's velocity \(t\) seconds after passing through a fixed point \(A\) on the line is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The graph of \(v\) against \(t\) consists of 4 straight line segments (see diagram). The woman is at the point \(B\) when \(t = 60\). Find

1. the woman's acceleration for \(0 < t < 30\) and for \(30 < t < 40\),
2. the distance \(A B\),
3. the total distance walked by the woman.