Questions - Edexcel M4 - A-Level Maths

Show that $e = \tan ^ { 2 } \theta$.

Edexcel M4 2003 January Q1

A boy enters a large horizontal field and sees a friend 100 m due north. The friend is walking in an easterly direction at a constant speed of $0.75 \mathrm {~m} \mathrm {~s} ^ { - 1 }$. The boy can walk at a maximum speed of $1 \mathrm {~m} \mathrm {~s} ^ { - 1 }$.

Find the shortest time for the boy to intercept his friend and the bearing on which he must travel to achieve this.
(6)

Edexcel M4 2003 January Q2

2. Boat $A$ is sailing due cast at a constant speed of $10 \mathrm {~km} \mathrm {~h} ^ { - 1 }$. To an observer on $A$, the wind appears to be blowing from due south. A second boat $B$ is sailing due north at a constant speed of $14 \mathrm {~km} \mathrm {~h} ^ { - 1 }$. To an observer on $B$, the wind appears to be blowing from the south west. The velocity of the wind relative to the earth is constant and is the same for both boats. Find the velocity of the wind relative to the earth, stating its magnitude and direction.

Edexcel M4 2003 January Q3

3. A small pebble of mass $m$ is placed in a viscous liquid and sinks vertically from rest through the liquid. When the speed of the pebble is $v$ the magnitude of the resistance due to the liquid is modelled as $m k v ^ { 2 }$, where $k$ is a positive constant. Find the speed of the pebble after it has fallen a distance $D$ through the liquid.
\includegraphics[max width=\textwidth, alt={}, center]{618fdb9c-cc0b-4a80-a148-4311c908c94e-3_813_699_397_674} Figure 1 shows a uniform rod $A B$, of mass $m$ and length $4 a$, resting on a smooth fixed sphere of radius $a$. A light elastic string, of natural length $a$ and modulus of elasticity $\frac { 3 } { 4 } m g$, has one end attached to the lowest point $C$ of the sphere and the other end attached to $A$. The points $A$, $B$ and $C$ lie in a vertical plane with $\angle B A C = 2 \theta$, where $\theta < \frac { \pi } { 4 }$. Given that $A C$ is always horizontal,

show that the potential energy of the system is $$\frac { m g a } { 8 } \left( 16 \sin 2 \theta + 3 \cot ^ { 2 } \theta - 6 \cot \theta \right) + \text { constant } ,$$
show that there is a value of $\theta$ for which the system is in equilibrium such that $0.535 < \theta < 0.545$.
Determine whether this position of equilibrium is stable or unstable.

Edexcel M4 2003 January Q5

5. A particle $P$ moves in a straight line. At time $t$ seconds its displacement from a fixed point $O$ on the line is $x$ metres. The motion of $P$ is modelled by the differential equation $$\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + 2 \frac { \mathrm {~d} x } { \mathrm {~d} t } + 2 x = 12 \cos 2 t - 6 \sin 2 t$$ When $t = 0 , P$ is at rest at $O$.

Find, in terms of $t$, the displacement of $P$ from $O$.
Show that $P$ comes to instantaneous rest when $t = \frac { \pi } { 4 }$.
Find, in metres to 3 significant figures, the displacement of $P$ from $O$ when $t = \frac { \pi } { 4 }$.
Find the approximate period of the motion for large values of $t$. \section*{6.} \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{618fdb9c-cc0b-4a80-a148-4311c908c94e-5_534_923_388_541}
\end{figure} A small ball $Q$ of mass $2 m$ is at rest at the point $B$ on a smooth horizontal plane. A second small ball $P$ of mass $m$ is moving on the plane with speed $\frac { 13 } { 12 } u$ and collides with $Q$. Both the balls are smooth, uniform and of the same radius. The point $C$ is on a smooth vertical wall $W$ which is at a distance $d _ { 1 }$ from $B$, and $B C$ is perpendicular to $W$. A second smooth vertical wall is perpendicular to $W$ and at a distance $d _ { 2 }$ from $B$. Immediately before the collision occurs, the direction of motion of $P$ makes an angle $\alpha$ with $B C$, as shown in Fig. 2, where tan $\alpha = \frac { 5 } { 12 }$. The line of centres of $P$ and $Q$ is parallel to $B C$. After the collision $Q$ moves towards $C$ with speed $\frac { 3 } { 5 } u$.
Show that, after the collision, the velocity components of $P$ parallel and perpendicular to $C B$ are $\frac { 1 } { 5 } u$ and $\frac { 5 } { 12 } u$ respectively.
Find the coefficient of restitution between $P$ and $Q$.
Show that when $Q$ reaches $C , P$ is at a distance $\frac { 4 } { 3 } d _ { 1 }$ from $W$. For each collision between a ball and a wall the coefficient of restitution is $\frac { 1 } { 2 }$.
Given that the balls collide with each other again,
show that the time between the two collisions of the balls is $\frac { 15 d _ { 1 } } { u }$,
find the ratio $d _ { 1 } : d _ { 2 }$. \section*{END}

Edexcel M4 2004 January Q1

A particle $P$ of mass 3 kg moves in a straight line on a smooth horizontal plane. When the speed of $P$ is $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$, the resultant force acting on $P$ is a resistance to motion of magnitude $2 v \mathrm {~N}$. Find the distance moved by $P$ while slowing down from $5 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ to $2 \mathrm {~m} \mathrm {~s} ^ { - 1 }$.
(5)

\begin{figure}[h]

\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{08aefb06-102f-4a1b-ae3c-2d0079b86045-2_731_1662_554_227}

\end{figure} Two smooth uniform spheres $A$ and $B$ of equal radius have masses 2 kg and 1 kg respectively. They are moving on a smooth horizontal plane when they collide. Immediately before the collision the speed of $A$ is $2.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ and the speed of $B$ is $1.3 \mathrm {~m} \mathrm {~s} ^ { - 1 }$. When they collide the line joining their centres makes an angle $\alpha$ with the direction of motion of $A$ and an angle $\beta$ with the direction of motion of $B$, where $\tan \alpha = \frac { 4 } { 3 }$ and $\tan \beta = \frac { 12 } { 5 }$ as shown in Fig. 1.

Find the components of the velocities of $A$ and $B$ perpendicular and parallel to the line of centres immediately before the collision. The coefficient of restitution between $A$ and $B$ is $\frac { 1 } { 2 }$.
Find, to one decimal place, the speed of each sphere after the collision.
(9)

Edexcel M4 2004 January Q3

3. \begin{figure}[h]

\captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{08aefb06-102f-4a1b-ae3c-2d0079b86045-3_933_1063_277_534}

\end{figure} Two uniform rods $A B$ and $A C$, each of mass $2 m$ and length $2 L$, are freely jointed at $A$. The mid-points of the rods are $D$ and $E$ respectively. A light inextensible string of length $s$ is fixed to $E$ and passes round small, smooth light pulleys at $D$ and $A$. A particle $P$ of mass $m$ is attached to the other end of the string and hangs vertically. The points $A , B$ and $C$ lie in the same vertical plane with $B$ and $C$ on a smooth horizontal surface. The angles $P A B$ and $P A C$ are each equal to $\theta ( \theta > 0 )$, as shown in Fig. 2.

Find the length of $A P$ in terms of $s , L$ and $\theta$.
Show that the potential energy $V$ of the system is given by $$V = 2 m g L ( 3 \cos \theta + \sin \theta ) + \text { constant } .$$
Hence find the value of $\theta$ for which the system is in equilibrium.
Determine whether this position of equilibrium is stable or unstable.

Edexcel M4 2004 January Q4

4. A particle $P$ of mass $m$ is attached to the mid-point of a light elastic string, of natural length $2 L$ and modulus of elasticity $2 m k ^ { 2 } L$, where $k$ is a positive constant. The ends of the string are attached to points $A$ and $B$ on a smooth horizontal surface, where $A B = 3 L$. The particle is released from rest at the point $C$, where $A C = 2 L$ and $A C B$ is a straight line. During the subsequent motion $P$ experiences air resistance of magnitude $2 m k v$, where $v$ is the speed of $P$. At time $t , A P = 1.5 L + x$.

Show that $\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + 2 k \frac { \mathrm {~d} x } { \mathrm {~d} t } + 4 k ^ { 2 } x = 0$.
Find an expression, in terms of $t , k$ and $L$, for the distance $A P$ at time $t$.

Edexcel M4 2004 January Q5

5. \begin{figure}[h]

\captionsetup{labelformat=empty} \caption{Figure 3} \includegraphics[alt={},max width=\textwidth]{08aefb06-102f-4a1b-ae3c-2d0079b86045-4_329_686_999_610}

\end{figure} Figure 3 represents the scene of a road accident. A car of mass 600 kg collided at the point $X$ with a stationary van of mass 800 kg . After the collision the van came to rest at the point $A$ having travelled a horizontal distance of 45 m , and the car came to rest at the point $B$ having travelled a horizontal distance of 21 m . The angle $A X B$ is $90 ^ { \circ }$. The accident investigators are trying to establish the speed of the car before the collision and they model both vehicles as small spheres.

Find the coefficient of restitution between the car and the van.
(5) The investigators assume that after the collision, and until the vehicles came to rest, the van was subject to a constant horizontal force of 500 N acting along $A X$ and the car to a constant horizontal force of 300 N along $B X$.
Find the speed of the car immediately before the collision.
(9)

Edexcel M4 2004 January Q6

6. \begin{figure}[h]

\captionsetup{labelformat=empty} \caption{Figure 4} \includegraphics[alt={},max width=\textwidth]{08aefb06-102f-4a1b-ae3c-2d0079b86045-5_431_1090_369_455}

\end{figure} Mary swims in still water at $0.85 \mathrm {~m} \mathrm {~s} ^ { - 1 }$. She swims across a straight river which is 60 m wide and flowing at $0.4 \mathrm {~m} \mathrm {~s} ^ { - 1 }$. She sets off from a point $A$ on the near bank and lands at a point $B$, which is directly opposite $A$ on the far bank, as shown in Fig. 4. Find

the angle between the near bank and the direction in which Mary swims,
the time she takes to cross the river. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 5} \includegraphics[alt={},max width=\textwidth]{08aefb06-102f-4a1b-ae3c-2d0079b86045-5_426_1290_1497_330}
\end{figure} A little further downstream a large tree has fallen from the far bank into the river. The river is modelled as flowing at $0.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ for a width of 40 m from the near bank, and $0.2 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ for the 20 m beyond this. Nassim swims at $0.85 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ in still water. He swims across the river from a point $C$ on the near bank. The point $D$ on the far bank is directly opposite $C$, as shown in Fig. 5. Nassim swims at the same angle to the near bank as Mary.
Find the maximum distance, downstream from CD, of Nassim during the crossing.
Show that he will land at the point $D$. \section*{END}

Edexcel M4 2005 January Q2

2. [In this question $\mathbf { i }$ and $\mathbf { j }$ are horizontal unit vectors due east and due north respectively.] A man cycling at a constant speed $u$ on horizontal ground finds that, when his velocity is $u \mathbf { j } \mathrm {~m} \mathrm {~s} ^ { - 1 }$, the velocity of the wind appears to be $v ( 3 \mathbf { i } - 4 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }$, where $v$ is a constant. When the velocity of the man is $\frac { u } { 5 } ( - 3 \mathbf { i } + 4 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }$, he finds that the velocity of the wind appears to be $w \mathbf { i } \mathrm {~m} \mathrm {~s} ^ { - 1 }$, where $w$ is a constant.

Show that $v = \frac { u } { 20 }$, and find $w$ in terms of $u$.
Find, in terms of $u$, the true velocity of the wind.

Edexcel M4 2005 January Q3

3. Two ships $A$ and $B$ are sailing in the same direction at constant speeds of $12 \mathrm {~km} \mathrm {~h} ^ { - 1 }$ and $16 \mathrm {~km} \mathrm {~h} ^ { - 1 }$ respectively. They are sailing along parallel lines which are 4 km apart. When the distance between the ships is $4 \mathrm {~km} , B$ turns through $30 ^ { \circ }$ towards $A$. Find the shortest distance between the ships in the subsequent motion.

Edexcel M4 2005 January Q4

4. A car of mass $M$ moves along a straight horizontal road. The total resistance to motion of the car is modelled as having constant magnitude $R$. The engine of the car works at a constant rate $R U$. Find the time taken for the car to accelerate from a speed of $\frac { 1 } { 4 } U$ to a speed of $\frac { 1 } { 2 } U$. \section*{5. [In this question $\mathbf { i }$ and $\mathbf { j }$ are horizontal perpendicular unit vectors.]} The vector $\mathbf { n } = \left( - \frac { 3 } { 5 } \mathbf { i } + \frac { 4 } { 5 } \mathbf { j } \right)$ and the vector $\mathbf { p } = \left( - \frac { 4 } { 5 } \mathbf { i } + \frac { 3 } { 5 } \mathbf { j } \right)$ are perpendicular unit vectors.

Verify that $\frac { 9 } { 5 } \mathbf { n } + \frac { 13 } { 5 } \mathbf { p } = ( \mathbf { i } + 3 \mathbf { j } )$. A smooth uniform sphere $S$ of mass 0.5 kg is moving on a smooth horizontal plane when it collides with a fixed smooth vertical wall which is parallel to $\mathbf { p }$. Immediately after the collision the velocity of $S$ is $( \mathbf { i } + 3 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }$. The coefficient of restitution between $S$ and the wall is $\frac { 9 } { 16 }$.
Find, in terms of $\mathbf { i }$ and $\mathbf { j }$, the velocity of $S$ immediately before the collision.
Find the energy lost in the collision. \section*{6.} \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{d6e5bd56-0a01-44a2-b439-f80cb356d46d-3_681_747_1121_679}
\end{figure} A smooth wire $P M Q$ is in the shape of a semicircle with centre $O$ and radius $a$. The wire is fixed in a vertical plane with $P Q$ horizontal and the mid-point $M$ of the wire vertically below $O$. A smooth bead $B$ of mass $m$ is threaded on the wire and is attached to one end of a light elastic string. The string has modulus of elasticity $4 m g$ and natural length $\frac { 5 } { 4 } a$. The other end of the string is attached to a fixed point $F$ which is a distance $a$ vertically above $O$, as shown in Fig. 1.

Edexcel M4 2005 January Q7

7. A particle of mass $m$ is attached to one end $P$ of a light elastic spring $P Q$, of natural length $a$ and modulus of elasticity $m a n ^ { 2 }$. At time $t = 0$, the particle and the spring are at rest on a smooth horizontal table, with the spring straight but unstretched and uncompressed. The end $Q$ of the spring is then moved in a straight line, in the direction $P Q$, with constant acceleration $f$. At time $t$, the displacement of the particle in the direction $P Q$ from its initial position is $x$ and the length of the spring is $( a + y )$.

Show that $x + y = \frac { 1 } { 2 } f t ^ { 2 }$.
Hence show that $$\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + n ^ { 2 } x = \frac { 1 } { 2 } n ^ { 2 } f t ^ { 2 }$$ You are given that the general solution of this differential equation is $$x = A \cos n t + B \sin n t + \frac { 1 } { 2 } f t ^ { 2 } - \frac { f } { n ^ { 2 } }$$ where $A$ and $B$ are constants.
Find the values of $A$ and $B$.
Find the maximum tension in the spring. END

Edexcel M4 2006 January Q1

A particle $P$ of mass 0.5 kg is released from rest at time $t = 0$ and falls vertically through a liquid. The motion of $P$ is resisted by a force of magnitude $2 v \mathrm {~N}$, where $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$ is the speed of $v$ at time $t$ seconds.
1. Show that $5 \frac { \mathrm {~d} v } { \mathrm {~d} t } = 49 - 20 v$.
2. Find the speed of $P$ when $t = 1$.
  (5)
3. A small smooth sphere $S$ of mass $m$ is attached to one end of a light inextensible string of length $2 a$. The other end of the string is attached to a fixed point $A$ which is at a distance $a \sqrt { } 3$ from a smooth vertical wall. The sphere $S$ hangs at rest in equilibrium. It is then projected horizontally towards the wall with a speed $\sqrt { \left( \frac { 37 g a } { 5 } \right) }$.
4. Show that $S$ strikes the wall with speed $\sqrt { \left( \frac { 27 g a } { 5 } \right) }$.
  (4)
Given that the loss in kinetic energy due to the impact with the wall is $\frac { 3 m g a } { 5 }$,
find the coefficient of restitution between $S$ and the wall.
(7)

Edexcel M4 2006 January Q3

3. Two ships $P$ and $Q$ are moving with constant velocity. At 3 p.m., $P$ is 20 km due north of $Q$ and is moving at $16 \mathrm {~km} \mathrm {~h} ^ { - 1 }$ due west. To an observer on ship $P$, ship $Q$ appears to be moving on a bearing of $030 ^ { \circ }$ at $10 \mathrm {~km} \mathrm {~h} ^ { - 1 }$. Find

1. the speed of $Q$,
2. the direction in which $Q$ is moving, giving your answer as a bearing to the nearest degree,
the shortest distance between the ships,
the time at which the two ships are closest together.
(3)

Edexcel M4 2006 January Q4

4. A particle $P$ of mass $m$ is suspended from a fixed point by a light elastic spring. The spring has natural length $a$ and modulus of elasticity $2 m \omega ^ { 2 } a$, where $\omega$ is a positive constant. At time $t = 0$ the particle is projected vertically downwards with speed $U$ from its equilibrium position. The motion of the particle is resisted by a force of magnitude $2 m \omega v$, where $v$ is the speed of the particle. At time $t$, the displacement of $P$ downwards from its equilibrium position is $x$.

Show that $\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + 2 \omega \frac { \mathrm {~d} x } { \mathrm {~d} t } + 2 \omega ^ { 2 } x = 0$. Given that the solution of this differential equation is $x = \mathrm { e } ^ { - \omega t } ( A \cos \omega t + B \sin \omega t )$, where $A$ and $B$ are constants,
find $A$ and $B$.
Find an expression for the time at which $P$ first comes to rest.
(3)

Edexcel M4 2006 January Q5

5. Two smooth uniform spheres $A$ and $B$ have equal radii. Sphere $A$ has mass $m$ and sphere $B$ has mass $k m$. The spheres are at rest on a smooth horizontal table. Sphere $A$ is then projected along the table with speed $u$ and collides with $B$. Immediately before the collision, the direction of motion of $A$ makes an angle of $60 ^ { \circ }$ with the line joining the centres of the two spheres. The coefficient of restitution between the spheres is $\frac { 1 } { 2 }$.

Show that the speed of $B$ immediately after the collision is $\frac { 3 u } { 4 ( k + 1 ) }$.
(6) Immediately after the collision the direction of motion of $A$ makes an angle arctan $( 2 \sqrt { 3 } )$ with the direction of motion of $B$.
Show that $k = \frac { 1 } { 2 }$.
Find the loss of kinetic energy due to the collision.
(4) \section*{6.} \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{fe647e21-9c4f-4035-b28f-b12a00087692-4_515_1077_301_502}
\end{figure} A smooth wire with ends $A$ and $B$ is in the shape of a semi-circle of radius $a$. The mid-point of $A B$ is $O$. The wire is fixed in a vertical plane and hangs below $A B$ which is horizontal. A small ring $R$, of mass $m \sqrt { 2 }$, is threaded on the wire and is attached to two light inextensible strings. The other end of each string is attached to a particle of mass $\frac { 3 m } { 2 }$. The particles hang vertically under gravity, as shown in Figure 1.
Show that, when the radius $O R$ makes an angle $2 \theta$ with the vertical, the potential energy, $V$, of the system is given by $$V = \sqrt { } 2 m g a ( 3 \cos \theta - \cos 2 \theta ) + \text { constant }$$
Find the values of $\theta$ for which the system is in equilibrium.
Determine the stability of the position of equilibrium for which $\theta > 0$.

Edexcel M4 2002 June Q1

1. \begin{figure}[h]

\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{c68c85a1-9d80-4ced-bfb6-c7b5347e9bb8-2_450_1417_391_339}

\end{figure} Two smooth uniform spheres $A$ and $B$, of equal radius, are moving on a smooth horizontal plane. Sphere $A$ has mass 2 kg and sphere $B$ has mass 3 kg . The spheres collide and at the instant of collision the line joining their centres is parallel to $\mathbf { i }$. Before the collision $A$ has velocity ( $3 \mathbf { i } - \mathbf { j }$ ) $\mathrm { m } \mathrm { s } ^ { - 1 }$ and after the collision it has velocity $( - 2 \mathbf { i } - \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }$. Before the collision the velocity of $B$ makes an angle $\alpha$ with the line of centres, as shown in Fig. 1, where $\tan \alpha = 2$. The coefficient of restitution between the spheres is $\frac { 1 } { 2 }$. Find, in terms of $\mathbf { i }$ and $\mathbf { j }$, the velocity of $B$ before the collision.
(9)

Edexcel M4 2002 June Q2

2. Ship $A$ is steaming on a bearing of $060 ^ { \circ }$ at $30 \mathrm {~km} \mathrm {~h} ^ { - 1 }$ and at 9 a.m. it is 20 km due west of a second ship $B$. Ship $B$ steams in a straight line.

Find the least speed of $B$ if it is to intercept $A$. Given that the speed of $B$ is $24 \mathrm {~km} \mathrm {~h} ^ { - 1 }$,
find the earliest time at which it can intercept $A$.

Edexcel M4 2002 June Q3

3. The engine of a car of mass 800 kg works at a constant rate of 32 kW . The car travels along a straight horizontal road and the resistance to motion of the car is proportional to the speed of the car. The car starts from rest and $t$ seconds later it has a speed of $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$.

Show that $$800 v \frac { \mathrm {~d} v } { \mathrm {~d} t } = 32000 - k v ^ { 2 } , \text { where } k \text { is a positive constant. }$$ Given that the limiting speed of the car is $40 \mathrm {~m} \mathrm {~s} ^ { - 1 }$, find
the value of $k$,
$v$ in terms of $t$.

Edexcel M4 2002 June Q4

4. \begin{figure}[h]

\captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{c68c85a1-9d80-4ced-bfb6-c7b5347e9bb8-3_424_422_1181_844}

\end{figure} Four identical uniform rods, each of mass $m$ and length $2 a$, are freely jointed to form a rhombus $A B C D$. The rhombus is suspended from $A$ and is prevented from collapsing by an elastic string which joins $A$ to $C$, with $\angle B A D = 2 \theta , 0 \leq \theta \leq \frac { 1 } { 3 } \pi$, as shown in Fig. 2. The natural length of the elastic string is $2 a$ and its modulus of elasticity is $4 m g$.

Show that the potential energy, $V$, of the system is given by $$V = 4 m g a \left[ ( 2 \cos \theta - 1 ) ^ { 2 } - 2 \cos \theta \right] + \text { constant } .$$
Hence find the non-zero value of $\theta$ for which the system is in equilibrium.
Determine whether this position of equilibrium is stable or unstable.

Edexcel M4 2002 June Q5

5. At time $t = 0$ particles $P$ and $Q$ start simultaneously from points which have position vectors $( \mathbf { i } - 2 \mathbf { j } + 3 \mathbf { k } ) \mathrm { m }$ and $( - \mathbf { i } + 2 \mathbf { j } - \mathbf { k } ) \mathrm { m }$ respectively, relative to a fixed origin $O$. The velocities of $P$ and $Q$ are $( \mathbf { i } + 2 \mathbf { j } - \mathbf { k } ) \mathrm { m } \mathrm { s } ^ { - 1 }$ and $( 2 \mathbf { i } + \mathbf { k } ) \mathrm { m } \mathrm { s } ^ { - 1 }$ respectively.

Show that $P$ and $Q$ collide and find the position vector of the point at which they collide. A third particle $R$ moves in such a way that its velocity relative to $P$ is parallel to the vector ( $- 5 \mathbf { i } + 4 \mathbf { j } - \mathbf { k }$ ) and its velocity relative to $Q$ is parallel to the vector $( - 2 \mathbf { i } + 2 \mathbf { j } - \mathbf { k } )$. Given that all three particles collide simultaneously, find
1. the velocity of $R$,
2. the position vector of $R$ at time $t = 0$.

Edexcel M4 2002 June Q6

6. \begin{figure}[h]

\captionsetup{labelformat=empty} \caption{Figure 3} \includegraphics[alt={},max width=\textwidth]{c68c85a1-9d80-4ced-bfb6-c7b5347e9bb8-4_244_1264_1314_382}

\end{figure} A particle $P$ of mass 2 kg is attached to the mid-point of a light elastic spring of natural length 2 m and modulus of elasticity 4 N . One end $A$ of the elastic spring is attached to a fixed point on a smooth horizontal table. The spring is then stretched until its length is 4 m and its other end $B$ is held at a point on the table where $A B = 4 \mathrm {~m}$. At time $t = 0 , P$ is at rest on the table at the point $O$ where $A O = 2 \mathrm {~m}$, as shown in Fig. 3. The end $B$ is now moved on the table in such a way that $A O B$ remains a straight line. At time $t$ seconds, $A B = \left( 4 + \frac { 1 } { 2 } \sin 4 t \right) \mathrm { m }$ and $A P = ( 2 + x ) \mathrm { m }$.

Show that $$\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + 4 x = \sin 4 t$$
Hence find the time when $P$ first comes to instantaneous rest. END

Edexcel M4 2003 June Q1

A wooden ball of mass 0.01 kg falls vertically into a pond of water. The speed of the ball as it enters the water is $10 \mathrm {~m} \mathrm {~s} ^ { - 1 }$. When the ball is $x$ metres below the surface of the water and moving downwards with speed $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$, the water provides a resistance of magnitude $0.02 v ^ { 2 } \mathrm {~N}$ and an upward buoyancy force of magnitude 0.158 N .
1. Show that, while the ball is moving downwards,
$$- 2 v ^ { 2 } - 6 = v \frac { \mathrm {~d} v } { \mathrm {~d} x }$$
Hence find, to 3 significant figures, the greatest distance below the surface of the water reached by the ball.

Questions — Edexcel M4 (159 questions)

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Edexcel M4 2004 January Q6

Edexcel M4 2005 January Q2

Edexcel M4 2005 January Q3

Edexcel M4 2005 January Q4

Edexcel M4 2005 January Q7

Edexcel M4 2006 January Q1

Edexcel M4 2006 January Q3

Edexcel M4 2006 January Q4

Edexcel M4 2006 January Q5

Edexcel M4 2002 June Q1

Edexcel M4 2002 June Q2

Edexcel M4 2002 June Q3

Edexcel M4 2002 June Q4

Edexcel M4 2002 June Q5

Edexcel M4 2002 June Q6

Edexcel M4 2003 June Q1