Variable acceleration (1D)

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 \(t\).

2 A particle moves in a straight line. Its velocity \(t\) seconds after leaving a fixed point \(O\) on the line is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), where \(v = 0.2 t + 0.006 t ^ { 2 }\). For the instant when the acceleration of the particle is 2.5 times its initial acceleration,

1. show that \(t = 25\),
2. find the displacement of the particle from \(O\).

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)

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 \(t = 5.5\).
[0pt]

2 A particle moves along the \(x\)-axis with velocity, \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), at time \(t\) given by $$v = 24 t - 6 t ^ { 2 }$$ The positive direction is in the sense of \(x\) increasing.

1. Find an expression for the acceleration of the particle at time \(t\).
2. Find the times, \(t _ { 1 }\) and \(t _ { 2 }\), at which the particle has zero speed.
3. Find the distance travelled between the times \(t _ { 1 }\) and \(t _ { 2 }\).

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 \(t = 5.5\).
[0pt]

A particle \(P\) moves in a straight line. It starts from rest at a point \(O\) on the line and at time \(t\) s after leaving \(O\) it has acceleration \(a \text{ m s}^{-2}\), where \(a = 6t - 18\). Find the distance \(P\) moves before it comes to instantaneous rest. [6]

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 \(k\).

6 A cyclist starts from rest at a fixed point \(O\) and moves in a straight line, before coming to rest \(k\) seconds later. The acceleration of the cyclist at time \(t \mathrm {~s}\) after leaving \(O\) is \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\), where \(a = 2 t ^ { - \frac { 1 } { 2 } } - \frac { 3 } { 5 } t ^ { \frac { 1 } { 2 } }\) for \(0 < t \leqslant k\).

1. Find the value of \(k\).
2. Find the maximum speed of the cyclist.
3. Find an expression for the displacement from \(O\) in terms of \(t\). Hence find the total distance travelled by the cyclist from the time at which she reaches her maximum speed until she comes to rest.

A particle \(P\) moves in a straight line so that, at time \(t\) seconds, its acceleration \(a\) m s\(^{-2}\) is given by $$a = \begin{cases} 4t - t^2, & 0 \leq t \leq 3, \\ \frac{27}{t^2}, & t > 3. \end{cases}$$ At \(t = 0\), \(P\) is at rest. Find the speed of \(P\) when

1. \(t = 3\), [3]
2. \(t = 6\). [5]

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.

A particle \(P\) moves on the \(x\)-axis. At time \(t\) seconds, its acceleration is \((5 - 2t)\) m s\(^{-2}\), measured in the direction of \(x\) increasing. When \(t = 0\), its velocity is 6 m s\(^{-1}\) measured in the direction of \(x\) increasing. Find the time when \(P\) is instantaneously at rest in the subsequent motion. [6]

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.

3 A particle \(P\) of mass 0.2 kg is released from rest and falls vertically. At time \(t \mathrm {~s}\) after release \(P\) has speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). A resisting force of magnitude \(0.8 v \mathrm {~N}\) acts on \(P\).

1. Show that the acceleration of \(P\) is \(( 10 - 4 v ) \mathrm { m } \mathrm { s } ^ { - 2 }\).
2. Find the value of \(v\) when \(t = 0.6\).

At time \(t\) seconds, a particle \(P\) has position vector \(r\) metres relative to a fixed origin \(O\), where $$r = (t^2 + 2t)\mathbf{i} + (t - 2t^2)\mathbf{j}.$$ Show that the acceleration of \(P\) is constant and find its magnitude. [5]

A particle moves in a circular path so that at time \(t\) seconds its position vector, \(\mathbf{r}\) metres, is given by $$\mathbf{r} = 4\sin(2t)\mathbf{i} + 4\cos(2t)\mathbf{j}$$ Find the velocity of the particle, in m s\(^{-1}\), when \(t = 0\) Circle your answer. [1 mark] \(8\mathbf{i}\) \quad \(-8\mathbf{j}\) \quad \(8\mathbf{j}\) \quad \(8\mathbf{i} - 8\mathbf{j}\)

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 \(t = 1\).

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 \(t = 8\).

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.

6 A particle moves in a straight line, starting from a point \(O\). The velocity of the particle at time \(t\) s after leaving \(O\) is \(v \mathrm {~ms} ^ { - 1 }\). It is given that \(\mathbf { v } = \mathrm { kt } ^ { \frac { 1 } { 2 } } - 2 \mathrm { t } - 8\), where \(k\) is a positive constant. The maximum velocity of the particle is \(4.5 \mathrm {~ms} ^ { - 1 }\).

1. Show that \(k = 10\).
2. 1. Verify that \(v = 0\) when \(t = 1\) and \(t = 16\).
   2. Find the distance travelled by the particle in the first 16 s .

A particle \(P\) of mass 1 kg is moving along a straight line against a resistive force of magnitude \(\frac{10\sqrt{v}}{(t+1)^2}\) N, where \(v\) ms\(^{-1}\) is the speed of \(P\) at time \(t\)s. When \(t = 0\), \(v = 25\). Find an expression for \(v\) in terms of \(t\). [5]

A particle \(P\) of mass 2 kg moves in a straight line along a smooth horizontal plane. The only horizontal force acting on \(P\) is a resistance of magnitude \(4v\) N, where \(v\) m s\(^{-1}\) is its speed. At time \(t = 0\) s, \(P\) has a speed of 5 m s\(^{-1}\). Find \(v\) in terms of \(t\). [6]

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.

A particle moves in a straight line starting from rest. The displacement \(s\) m of the particle from a fixed point \(O\) on the line at time \(t\) s is given by $$s = t^2 - \frac{15}{4}t^2 + 6.$$ Find the value of \(s\) when the particle is again at rest. [4]

5 A particle \(P\) moves in a straight line. It starts at a point \(O\) on the line and at time \(t\) s after leaving \(O\) it has velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), where \(v = 4 t ^ { 2 } - 20 t + 21\).

1. Find the values of \(t\) for which \(P\) is at instantaneous rest.
2. Find the initial acceleration of \(P\).
3. Find the minimum velocity of \(P\).
4. Find the distance travelled by \(P\) during the time when its velocity is negative.

6 A particle travels in a straight line from a point \(P\) to a point \(Q\). Its velocity \(t\) seconds after leaving \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), where \(v = 4 t - \frac { 1 } { 16 } t ^ { 3 }\). The distance \(P Q\) is 64 m .

1. Find the time taken for the particle to travel from \(P\) to \(Q\).
2. Find the set of values of \(t\) for which the acceleration of the particle is positive.

6 Particle \(P\) travels in a straight line from \(A\) to \(B\). The velocity of \(P\) at time \(t \mathrm {~s}\) after leaving \(A\) is denoted by \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), where $$v = 0.04 t ^ { 3 } + c t ^ { 2 } + k t$$ \(P\) takes 5 s to travel from \(A\) to \(B\) and it reaches \(B\) with speed \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The distance \(A B\) is 25 m .

1. Find the values of the constants \(c\) and \(k\).
2. Show that the acceleration of \(P\) is a minimum when \(t = 2.5\).

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 \(t = 1.5\) .

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}

4 Water flows into a bowl at a constant rate of \(10 \mathrm {~cm} ^ { 3 } \mathrm {~s} ^ { - 1 }\) (see Fig. 4). \begin{figure}[h] \end{figure} When the depth of water in the bowl is \(h \mathrm {~cm}\), the volume of water is \(V \mathrm {~cm} ^ { 3 }\), where \(V = \pi h ^ { 2 }\). Find the rate at which the depth is increasing at the instant in time when the depth is 5 cm .

3 \begin{figure}[h] \end{figure} \begin{figure}[h] \end{figure} Fig. 1 shows an open tank in the shape of a triangular prism. The vertical ends \(A B E\) and \(D C F\) are identical isosceles triangles. Angle \(A B E =\) angle \(B A E = 30 ^ { \circ }\). The length of \(A D\) is 40 cm . The tank is fixed in position with the open top \(A B C D\) horizontal. Water is poured into the tank at a constant rate of \(200 \mathrm {~cm} ^ { 3 } \mathrm {~s} ^ { - 1 }\). The depth of water, \(t\) seconds after filling starts, is \(h \mathrm {~cm}\) (see Fig. 2).

1. Show that, when the depth of water in the tank is \(h \mathrm {~cm}\), the volume, \(V \mathrm {~cm} ^ { 3 }\), of water in the tank is given by \(V = ( 40 \sqrt { } 3 ) h ^ { 2 }\).
2. Find the rate at which \(h\) is increasing when \(h = 5\).

3 \begin{figure}[h] \end{figure} \begin{figure}[h] \end{figure} Fig. 1 shows an open tank in the shape of a triangular prism. The vertical ends \(A B E\) and \(D C F\) are identical isosceles triangles. Angle \(A B E =\) angle \(B A E = 30 ^ { \circ }\). The length of \(A D\) is 40 cm . The tank is fixed in position with the open top \(A B C D\) horizontal. Water is poured into the tank at a constant rate of \(200 \mathrm {~cm} ^ { 3 } \mathrm {~s} ^ { - 1 }\). The depth of water, \(t\) seconds after filling starts, is \(h \mathrm {~cm}\) (see Fig. 2).

1. Show that, when the depth of water in the tank is \(h \mathrm {~cm}\), the volume, \(V \mathrm {~cm} ^ { 3 }\), of water in the tank is given by \(V = ( 40 \sqrt { } 3 ) h ^ { 2 }\).
2. Find the rate at which \(h\) is increasing when \(h = 5\).

A particle \(P\) moves in a straight line so that its velocity \(v\) ms\(^{-1}\) at time \(t\) seconds is given, for \(t > 1\), by the formula \(v = 2t + \frac{8}{t^2}\). Find the time when the acceleration of \(P\) is zero. [5 marks]

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 \(t\).

3 A particle \(P\) moves along a straight line for 100 s . It starts at a point \(O\) and at time \(t\) seconds after leaving \(O\) the velocity of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), where $$v = 0.00004 t ^ { 3 } - 0.006 t ^ { 2 } + 0.288 t$$

1. Find the values of \(t\) at which the acceleration of \(P\) is zero.
2. Find the displacement of \(P\) from \(O\) when \(t = 100\).

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 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 \(V = 100\).

A particle \(P\) of mass \(2 \text{ kg}\) moving on a horizontal straight line has displacement \(x \text{ m}\) from a fixed point \(O\) on the line and velocity \(v \text{ m s}^{-1}\) at time \(t \text{ s}\). The only horizontal force acting on \(P\) is a variable force \(F \text{ N}\) which can be expressed as a function of \(t\). It is given that $$\frac{v}{x} = \frac{3-t}{1+t}$$ and when \(t = 0\), \(x = 5\).

1. Find an expression for \(x\) in terms of \(t\). [4]

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

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 \(t\).

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)

2 The volume of a spherical balloon is increasing at a constant rate of \(50 \mathrm {~cm} ^ { 3 }\) per second. Find the rate of increase of the radius when the radius is 10 cm . [Volume of a sphere \(= \frac { 4 } { 3 } \pi r ^ { 3 }\).]

2 The volume of a spherical balloon is increasing at a constant rate of \(50 \mathrm {~cm} ^ { 3 }\) per second. Find the rate of increase of the radius when the radius is 10 cm . [Volume of a sphere \(= \frac { 4 } { 3 } \pi r ^ { 3 }\).]

8. The volume \(V\) of a spherical balloon is increasing at a constant rate of \(250 \mathrm {~cm} ^ { 3 } \mathrm {~s} ^ { - 1 }\). Find the rate of increase of the radius of the balloon, in \(\mathrm { cm } \mathrm { s } ^ { - 1 }\), at the instant when the volume of the balloon is \(12000 \mathrm {~cm} ^ { 3 }\).
Give your answer to 2 significant figures.
[0pt] [You may assume that the volume \(V\) of a sphere of radius \(r\) is given by the formula \(V = \frac { 4 } { 3 } \pi r ^ { 3 }\).]

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.

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 \(t = 2\).

A particle \(P\) moves on the \(x\)-axis from the origin \(O\) with an initial velocity of \(-20\) m s\(^{-1}\). The acceleration \(a\) m s\(^{-2}\) at time \(t\) s after leaving \(O\) is given by \(a = 12 - 2t\).

1. Sketch a velocity-time graph for \(0 \leq t \leq 12\), indicating the times when \(P\) is at rest. [5]
2. Find the total distance travelled by \(P\) in the interval \(0 \leq t \leq 12\). [5]

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.

1 A curve passes through the point \(( 0,1 )\) and satisfies the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \sqrt { 1 + x ^ { 2 } }$$ Starting at the point \(( 0,1 )\), use a step-by-step method with a step length of 0.2 to estimate the value of \(y\) at \(x = 0.4\). Give your answer to five decimal places.

1. A particle \(P\) of mass 2 kg is subjected to a force \(\mathbf { F }\) such that its displacement, \(\mathbf { r }\) metres, from a fixed origin, \(O\), at time \(t\) seconds is given by

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

1. Show that the acceleration of \(P\) is constant.
2. Find the magnitude of \(\mathbf { F }\).

2. At time \(t\) seconds, where \(t \geqslant 0\), a particle \(P\) is moving on a horizontal plane with acceleration \(\left[ \left( 3 t ^ { 2 } - 4 t \right) \mathbf { i } + ( 6 t - 5 ) \mathbf { j } \right] \mathrm { m } \mathrm { s } ^ { - 2 }\). When \(t = 3\) the velocity of \(P\) is \(( 11 \mathbf { i } + 10 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\). Find

1. the velocity of \(P\) at time \(t\) seconds,
2. the speed of \(P\) when it is moving parallel to the vector \(\mathbf { i }\).

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 \(y\).

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}