Questions M1 - A-Level Maths

Find the value of $F$,
Find the magnitude of the force in the $\operatorname { rod } A B$.
State the modelling assumption that you have made about the lump of clay.
(6 marks)
(2 marks)
(1 mark)

Edexcel M1 Q4

4. Two particles $A$ and $B$, of masses 50 grams and $y$ grams, are moving in the same straight line, in opposite directions, with speeds $7 \mathrm {~ms} ^ { - 1 }$ and $4 \mathrm {~ms} ^ { - 1 }$ respectively, and collide.
\includegraphics[max width=\textwidth, alt={}, center]{31efa627-5114-4797-9d46-7f1311c18ff8-1_218_508_2143_1382}
In each of the following separate cases, find the value of $y$ and the magnitude of the impulse exerted by each particle on the other:

after impact the particles move together with speed $2.25 \mathrm {~ms} ^ { - 1 }$;
after impact the particles move in opposite directions with speed $5 \mathrm {~ms} ^ { - 1 }$. \section*{MECHANICS 1 (A) TEST PAPER 6 Page 2}

Edexcel M1 Q5

\includegraphics[max width=\textwidth, alt={}]{31efa627-5114-4797-9d46-7f1311c18ff8-2_262_597_276_356}

A small stone is projected with speed $7 \mathrm {~ms} ^ { - 1 }$ from $P$, the bottom of a rough plane inclined at $25 ^ { \circ }$ to the horizontal, and moves up a line of greatest slope of the plane until it comes to instantaneous rest at $Q$, where $P Q = 4 \mathrm {~m}$.

Show that the deceleration of the stone as it moves up the plane has magnitude $\frac { 49 } { 8 } \mathrm {~ms} ^ { - 2 }$.
Find the coefficient of friction between the stone and the plane,
Find the speed with which the stone returns to $P$.
Name one force which you have ignored in your mathematical model, and state whether the answer to part (c) would be larger or smaller if that force were taken into account.

Edexcel M1 Q6

6. The points $A$ and $B$ have position vectors $( 30 \mathbf { i } - 60 \mathbf { j } ) \mathrm { m }$ and $( - 20 \mathbf { i } + 60 \mathbf { j } ) \mathrm { m }$ respectively relative to an origin $O$, where $\mathbf { i }$ and $\mathbf { j }$ are perpendicular unit vectors. A cyclist, Chris, starts at $A$ and cycles towards $B$ with constant speed $2.6 \mathrm {~ms} ^ { - 1 }$. Another cyclist, Doug, starts at $O$ and cycles towards $B$ with constant speed $k \sqrt { } 10 \mathrm {~ms} ^ { - 1 }$.

Show that Chris's velocity vector is $( - \mathbf { i } + 2 \cdot 4 \mathbf { j } ) \mathrm { ms } ^ { - 1 }$.
Find Doug's velocity vector in the form $k ( a \mathbf { i } + b \mathbf { j } ) \mathrm { ms } ^ { - 1 }$. Given that Chris and Doug arrive at $B$ at the same time,
find the value of $k$.

Edexcel M1 Q7

7.
\includegraphics[max width=\textwidth, alt={}, center]{31efa627-5114-4797-9d46-7f1311c18ff8-2_456_380_1862_395} A particle $P$, of mass 4 kg , rests on horizontal ground and is attached by a light, inextensible string to another particle $Q$ of mass 4.5 kg . The string passes over a smooth pulley whose centre is 3 m above the ground. Initially $Q$ is 1.1 m below the level of the centre of the pulley. The system is released from rest in this position.

Find the acceleration of the two particles.
Find the speed with which $Q$ hits the ground. Assuming that $Q$ does not rebound from the ground while the string is slack,
show that $P$ does not reach the pulley before $Q$ starts to move again.
Find the speed with which $Q$ leaves the ground when the string again becomes taut.
(3 marks)

Edexcel M1 Q1

Briefly define the following terms used in modelling in Mechanics:
1. lamina,
2. uniform rod,
3. smooth surface,
4. particle.
  (4 marks)
5. Two forces $\mathbf { F }$ and $\mathbf { G }$ are given by $\mathbf { F } = ( 6 \mathbf { i } - 5 \mathbf { j } ) \mathbf { N } , \mathbf { G } = ( 3 \mathbf { i } + 17 \mathbf { j } ) \mathrm { N }$, where $\mathbf { i }$ and $\mathbf { j }$ are unit vectors in the $x$ and $y$ directions respectively and the unit of length on each axis is 1 cm .
  (a) Find the magnitude of $\mathbf { R }$, the resultant of $\mathbf { F }$ and $\mathbf { G }$.
  (b) Find the angle between the direction of $\mathbf { R }$ and the positive $x$-axis.
  $\mathbf { R }$ acts through the point $P ( - 4,3 )$. $O$ is the origin $( 0,0 )$.
  (c) Use the fact that $O P$ is perpendicular to the line of action of $\mathbf { R }$ to calculate the moment of $\mathbf { R }$ about an axis through the origin and perpendicular to the $x - y$ plane.
6. A string is attached to a packing case of mass 12 kg , which is at rest on a rough horizontal plane. When a force of magnitude 50 N is applied at the other end of the string, and
  \includegraphics[max width=\textwidth, alt={}, center]{6fbe12d6-9a46-4602-a7b9-63d50b02ff28-1_183_522_1106_1421}
  the string makes an angle of $35 ^ { \circ }$ with the vertical as shown, the case is on the point of moving.
  (a) Find the coefficient of friction between the case and the plane.
The force is now increased, with the string at the same angle, and the case starts to move along the plane with constant acceleration, reaching a speed of $2 \mathrm {~ms} ^ { - 1 }$ after 4 seconds.
(b) Find the magnitude of the new force.
(c) State any modelling assumptions you have made about the case and the string.

Edexcel M1 Q4

4. A uniform yoke $A B$, of mass 4 kg and length $4 a \mathrm {~m}$, rests on the shoulders $S$ and $T$ of two oxen.
\includegraphics[max width=\textwidth, alt={}, center]{6fbe12d6-9a46-4602-a7b9-63d50b02ff28-1_131_679_1800_1275}
$A S = T B = a \mathrm {~m}$. A bucket of mass $x \mathrm {~kg}$ is suspended from $A$.

Show that the vertical force on the yoke at $T$ has magnitude $\left( 2 - \frac { 1 } { 2 } x \right) g \mathrm {~N}$ and find, in terms of $x$ and $g$, the vertical force on the yoke at $S$.
If the ratio of these vertical forces is $5 : 1$, find the value of $x$.
Find the maximum value of $x$ for which the yoke will remain horizontal.

Edexcel M1 Q5

5. Two small smooth spheres $A$ and $B$, of equal radius but masses $m \mathrm {~kg}$ and km kg respectively, where $k > 1$, move towards each other along a straight line and collide directly. Immediately before the collision, $A$ has speed $5 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ and $B$ has speed $3 \mathrm {~m} \mathrm {~s} ^ { - 1 }$. In the collision, the impulse exerted by $A$ on $B$ has magnitude 7 km Ns.
Q. 5 continued on next page ... \section*{MECHANICS 1 (A) TEST PAPER 7 Page 2}

continued ...
1. Find the speed of $B$ after the impact.
2. Show that the speed of $A$ immediately after the collision is $( 7 k - 5 ) \mathrm { ms } ^ { - 1 }$ and deduce that the direction of $A$ 's motion is reversed.
  $B$ is now given a further impulse of magnitude $m u \mathrm { Ns }$, as a result of which a second collision between it and $A$ occurs.
3. Show that $u > k ( 7 k - 1 )$.
\includegraphics[max width=\textwidth, alt={}]{6fbe12d6-9a46-4602-a7b9-63d50b02ff28-2_422_787_815_340}
The velocity-time graph illustrates the motion of a particle which accelerates from rest to $8 \mathrm {~ms} ^ { - 1 }$ in $x$ seconds and then to $24 \mathrm {~ms} ^ { - 1 }$ in a further 4 seconds. It then travels at a constant speed for another $y$ seconds before decelerating to $12 \mathrm {~ms} ^ { - 1 }$ over the next $y$ seconds and then to rest in the final 7 seconds of its motion. Given that the total distance travelled by the particle is 496 m ,
show that $2 x + 21 y = 195$. Given also that the average speed of the particle during its motion is $15 \cdot 5 \mathrm {~ms} ^ { - 1 }$,
show that $x + 2 y = 21$.
Hence find the values of $x$ and $y$,
Write down the acceleration for each section of the motion.

Edexcel M1 Q7

7. Two particles $P$ and $Q$, of masses $2 m$ and $3 m$ respectively, are connected by a light string. Initially, $P$ is at rest on a smooth horizontal table. The string passes over a small smooth pulley and $Q$ rests on a rough plane inclined at an
\includegraphics[max width=\textwidth, alt={}, center]{6fbe12d6-9a46-4602-a7b9-63d50b02ff28-2_241_483_1818_1407}
angle $\theta$ to the horizontal, where $\tan \theta = \frac { 4 } { 3 }$. The coefficient of friction between $Q$ and the inclined plane is $\frac { 1 } { 6 }$.
The system is released from rest with $Q$ at a distance of 0.8 metres above a horizontal floor.

Show that the acceleration of $P$ and $Q$ is $\frac { 21 g } { 50 }$, stating a modelling assumption which you must make to ensure that both particles have the same acceleration.
Find the speed with which $Q$ hits the floor. After $Q$ hits the floor and does not rebound, $P$ moves a further 0.2 m until it hits the pulley.
Find the total time after the system is released before $P$ hits the pulley.

Edexcel M1 Q1

A golf ball and a table tennis ball are dropped together from the top of a building. The golf ball hits the ground after 1.7 seconds.
1. Calculate the height of the top of the building above the ground.
According to a simple model, the two balls hit the ground at the same time.
State why this may not be true in practice and describe a refinement to the model which could lead to a more realistic solution.

Edexcel M1 Q2

2. A plank of wood $X Y$ has length $5 a$ m and mass 5 kg . It rests on a support at $Q$, where $X Q = 3 a$
m . When a kitten of mass 8 kg sits on the plank at $P$, where $P Y = a \mathrm {~m}$, the plank just remains horizontal. By modelling the plank as a non-uniform rod and the kitten as a particle, find

the magnitude of the reaction at the support,
the distance from $X$ to the centre of mass of the plank, in terms of $a$.

Edexcel M1 Q3

3. A particle is in equilibrium under the action of three forces $\mathbf { P } , \mathbf { Q }$ and $\mathbf { R }$ acting in the same horizontal plane. $P$ has magnitude 9 N and acts on a bearing of $030 ^ { \circ } . Q$ has magnitude 12 N . and acts on a bearing of $225 ^ { \circ }$.

Find the values of $a$ and $b$ such that $\mathbf { R } = ( a \mathbf { i } + b \mathbf { j } ) \mathrm { N }$, where $\mathbf { i }$ and $\mathbf { j }$ are unit vectors in the directions due East and due North respectively.
Calculate the magnitude and direction of $\mathbf { R }$

Edexcel M1 Q4

4. $X$ and $Y$ are two points 1 m apart on a line of greatest slope of a smooth plane inclined at $60 ^ { \circ }$ to the horizontal. A particle $P$ of mass 1 kg is released from rest at $X$.

Find the speed with which $P$ reaches $Y$.
$P$ is now connected to another particle $Q$, of mass $M \mathrm {~kg}$, by a light inextensible string. The system is placed with $P$ at $Y$ on the plane and $Q$ hanging vertically at the other end of the string, which passes over a fixed pulley at the top of the plane.
The system is released from rest and $P$ moves up the plane with acceleration $\frac { g } { 5 }$.
\includegraphics[max width=\textwidth, alt={}, center]{cc75a4a5-1c3a-4e36-acfd-21f6246f2a38-1_358_321_2024_1597}
Show that $M = \frac { 5 \sqrt { } 3 + 2 } { 8 }$.
State a modelling assumption that you have made about the pulley. Briefly state what would be implied if this assumption were not made. \section*{MECHANICS 1 (A) TEST PAPER 8 Page 2}

Edexcel M1 Q5

Two model cars $A$ and $B$ have masses 200 grams and $k$ grams respectively. They move towards each other in a straight line and collide directly when their speeds are $5 \mathrm {~ms} ^ { - 1 }$ and $4 \mathrm {~ms} ^ { - 1 }$ respectively. As a result the speed of $A$ is reduced to $2 \mathrm {~ms} ^ { - 1 }$, in the same direction as before. The direction of $B$ 's motion is reversed and its speed immediately after the impact is $5 \mathrm {~ms} ^ { - 1 }$.
1. Find the magnitude of the impulse exerted by $A$ on $B$ in the impact. State the units of your answer.
2. Find the value of $k$.
The surface on which the cars move is rough, and $B$ comes to rest 3 seconds after the impact. The coefficient of friction between both cars and the surface is $\mu$.
Find the value of $\mu$.
Find the distance travelled by $A$ after the impact before it comes to rest.

Edexcel M1 Q6

6. A small ring, of mass $m \mathrm {~kg}$, can slide along a straight wire which is fixed at an angle of $45 ^ { \circ }$ to the horizontal as shown. The coefficient of friction between the ring and the wire is $\frac { 2 } { 7 }$.
The ring rests in equilibrium on the wire and is just prevented from
\includegraphics[max width=\textwidth, alt={}, center]{cc75a4a5-1c3a-4e36-acfd-21f6246f2a38-2_273_296_1192_1617}
sliding down the wire when a horizontal string is attached to it, as shown

Show that the tension in the string has magnitude $\frac { 5 m g } { 9 } \mathrm {~N}$. The string is now removed and the ring starts to slide down the wire.
Find the time that elapses before the ring has moved 10 cm along the wire.

Edexcel M1 Q7

7. Two cyclists, Alice and Bobbie, travel from $P$ to $Q$ along a straight path. Alice starts from rest at $P$ just as Bobbie passes her at $3.5 \mathrm {~ms} ^ { - 1 }$. Bobbie continues at this speed while Alice accelerates at $0.2 \mathrm {~ms} ^ { - 2 }$ for $T$ seconds until she attains her maximum speed. At this moment both cyclists immediately start to slow down, with constant but different decelerations, and they come to rest at $Q 80$ seconds after Alice started moving.

Sketch, on the same diagram, the velocity-time graphs for the two cyclists. By using the fact that both cyclists cover the same distance, find
the value of $T$,
the distance between $P$ and $Q$,
the magnitude of Bobbie's deceleration.

Edexcel M1 Q1

A car accelerates from 0 to $108 \mathrm {~km} \mathrm {~h} ^ { - 1 }$ in 7.5 seconds. Find its acceleration in $\mathrm { ms } ^ { - 2 }$. ( 3 marks)
A book rests on a rough desk-lid which is hinged at one end. When the lid is raised so that it makes an angle of $15 ^ { \circ }$ with the horizontal, the book is just on the point of sliding down a line of greatest slope. Modelling the book as a particle, find
1. the coefficient of friction between the book and the desk-lid,
2. the acceleration with which the book starts to move if it is released from rest when the lid is inclined at $20 ^ { \circ }$ to the horizontal.
3. A particle $P$ is projected vertically upwards from ground level at time $t = 0$ with speed 20 $\mathrm { ms } ^ { - 1 }$. Two seconds later another particle $Q$ is projected vertically upwards with speed 30 $\mathrm { ms } ^ { - 1 }$ from a point on the same horizontal ground.
4. Taking the upward direction as positive, write down expressions in terms of $g$ and $t$ for the velocities of $P$ and of $Q$ at time $t$ seconds after $P$ is projected.
5. Find the value of $t$ when both particles are moving with the same speed.
6. A jet of water issues from a cylindrical pipe with a circular cross-section of radius $2 \cdot 75 \mathrm {~cm}$. The water strikes a vertical wall at a speed of $9 \mathrm {~ms} ^ { - 1 }$. Taking the density of water to be $1000 \mathrm {~kg} \mathrm {~m} ^ { - 3 }$, calculate
7. the momentum destroyed each second by the impact with the wall,
8. the magnitude of the force exerted by the water on the wall.
9. State one modelling assumption that you have made.
10. Two particles $A$ and $B$, of mass 1 kg and $m \mathrm {~kg}$ respectively, where $m > 1$, are attached to the ends of a light inextensible string which passes over a small fixed smooth pulley. The particles are released from rest and move with the string taut and vertical.
11. Show that the acceleration of the system is equal to $\frac { ( m - 1 ) g } { m + 1 }$.
12. Find the tension in the string, in terms of $m$ and $g$, expressing your answer as a single algebraic fraction in its simplest form.
When the system is released from rest, both particles are $52 \cdot 5 \mathrm {~cm}$ above ground level and 60 cm below the level of the pulley. $B$ hits the ground after half a second.
Find the value of $m$.
Find the speed with which $B$ hits the ground. \section*{MECHANICS 1 (A) TEST PAPER 9 Page 2}

Edexcel M1 Q6

At noon, two boats $P$ and $Q$ have position vectors $( \mathbf { i } + 7 \mathbf { j } ) \mathrm { km }$ and $( 3 \mathbf { i } - 8 \mathbf { j } ) \mathrm { km }$ respectively relative to an origin $O$, where $\mathbf { i }$ and $\mathbf { j }$ are unit vectors in the directions due East and due North respectively. $P$ is moving with constant velocity $( 3 \mathbf { i } - 4 \mathbf { j } ) \mathrm { km } \mathrm { h } ^ { - 1 }$ and $Q$ is moving with constant velocity $( 6 \mathbf { i } + 5 \mathbf { j } ) \mathrm { km } \mathrm { h } ^ { - 1 }$.
1. Find the position vector of each boat at time $t$ hours after noon, giving your answers in the form $\mathrm { f } ( t ) \mathrm { i } + \mathrm { g } ( t ) \mathrm { j }$, where $\mathrm { f } ( t )$ and $\mathrm { g } ( t )$ are linear functions of $t$ to be found.
2. Find, in terms of $t$, the distance between the boats $t$ hours after noon.
3. Calculate the time when the boats are closest together and find the distance between them at this time.
4. A particle starts from rest and accelerates at a uniform rate over a distance of 12 m . It then travels at a constant speed of $u \mathrm {~ms} ^ { - 1 }$ for a further 30 seconds. Finally it decelerates uniformly to rest at $1.6 \mathrm {~ms} ^ { - 2 }$.
5. Sketch the velocity-time graph for this motion.
6. Show that the total time for which the particle is in motion is
$$\frac { 5 u } { 8 } + 30 + \frac { 24 } { u } \text { seconds. }$$
Find, in terms of $u$, the total distance travelled by the particle during the motion.
Given that the total time for the motion is $39 \cdot 5$ seconds, show that $5 u ^ { 2 } - 76 u + 192 = 0$.
Find the two possible values of $u$ and the total distance travelled in each case.

Edexcel M1 Q1

A boy holds a 30 cm metal ruler between three fingers of one hand, pushing down with the middle finger and up with the other two, at the points marked $5 \mathrm {~cm} , 10 \mathrm {~cm}$
\includegraphics[max width=\textwidth, alt={}, center]{88b98fe3-7952-44e4-8a8e-5f6f8813fbcc-1_197_556_354_1371}
and $x \mathrm {~cm}$ and exerting forces of magnitude $11 \mathrm {~N} , 18 \mathrm {~N}$ and 8 N respectively. The ruler is in equilibrium in this position. Modelling the ruler as a uniform rod, find
1. the mass of the ruler, in grams,
2. the value of $x$.
3. State how you have used the modelling assumption that the ruler is a uniform rod.
\includegraphics[max width=\textwidth, alt={}, center]{88b98fe3-7952-44e4-8a8e-5f6f8813fbcc-1_182_372_1000_367}
A small packet of mass 0.3 kg rests on a rough horizontal surface. The coefficient of friction between the packet and the surface is $\frac { 1 } { 4 }$. Two strings are attached to the packet, making angles of $45 ^ { \circ }$ and $30 ^ { \circ }$ with the horizontal, and when forces of magnitude 2 N and $F \mathrm {~N}$ are exerted through the strings as shown, the packet is on the point of moving in the direction $\overrightarrow { A B }$.
Find the value of $F$.

Edexcel M1 Q3

3. A body moves in a straight line with constant acceleration. Its speed increases from $17 \mathrm {~ms} ^ { - 1 }$ to $33 \mathrm {~ms} ^ { - 1 }$ while it travels a distance of 250 m . Find

the time taken to travel the 250 m ,
the acceleration of the body. The body now decelerates at a constant rate from $33 \mathrm {~ms} ^ { - 1 }$ to rest in 6 seconds.
Find the distance travelled in these 6 seconds.

Edexcel M1 Q4

4. A particle $P$ of mass $m \mathrm {~kg}$, at rest on a smooth horizontal table, is connected to particles $Q$ and $R$, of mass 0.1 kg and 0.5 kg respectively, by strings which pass over fixed pulleys at the edges of the table. The system is released from rest with $Q$ and $R$ hanging freely and it is found that the tension in the section of the string between $P$ and $R$ is 2 N .

Show that the acceleration of the particles has magnitude $5 \cdot 8 \mathrm {~ms} ^ { - 2 }$.
Find the value of $m$. Modelling assumptions have been made about the pulley and the strings.
Briefly describe these two assumptions. For each one, state how the mathematical model would be altered if the assumption were not made. \section*{MECHANICS 1 (A)TEST PAPER 10 Page 2}

Edexcel M1 Q5

Two trucks $P$ and $Q$, of masses 18000 kg and 16000 kg respectively, collide while moving towards each other in a straight line. Immediately before the collision, both trucks are travelling at the same speed, $u \mathrm {~ms} ^ { - 1 }$. Immediately after the collision, $P$ is moving at half its original speed, its direction of motion having been reversed.
1. Find, in terms of $u$, the speed of $Q$ immediately after the collision.
2. State, with a reason, whether the direction of $Q$ 's motion has been reversed.
3. Find, in terms of $u$, the magnitude of the impulse exerted by $P$ on $Q$ in the collision, stating the units of your answer.
The force exerted by each truck on the other in the impact has magnitude $108000 u \mathrm {~N}$.
Find the time for which the trucks are in contact.

Edexcel M1 Q6

6. A particle $P$ moves in a straight line such that its displacement from a fixed point $O$ at time $t \mathrm {~s}$ is $y$ metres. The graph of $y$ against $t$ is as shown.

Write down the velocity of $P$ when
1. $t = 1$,
2. $t = 10$.
  (2 marks)
State the total distance travelled by $P$.
(2 marks)
Write down a formula for $y$ in terms of $t$ when $2 \leq t < 4$.
(3 marks)
\includegraphics[max width=\textwidth, alt={}, center]{88b98fe3-7952-44e4-8a8e-5f6f8813fbcc-2_634_771_1032_1192}
Sketch a velocity-time graph for the motion of $P$ during the twelve seconds.
Find the maximum speed of $P$ during the motion.

Edexcel M1 Q7

7. Two trains $S$ and $T$ are moving with constant speeds on straight tracks which intersect at the point $O$. At 9.00 a.m. $S$ has position vector $( - 10 \mathbf { i } + 24 \mathbf { j } ) \mathrm { km }$ and $T$ has position vector $25 \mathbf { j }$ km relative to $O$, where $\mathbf { i }$ and $\mathbf { j }$ are unit vectors in the directions due east and due north respectively. $S$ is moving with speed $52 \mathrm {~km} \mathrm {~h} ^ { - 1 }$ and $T$ is moving with speed $50 \mathrm {~km} \mathrm {~h} ^ { - 1 }$, both towards $O$.

Show that the velocity vector of $S$ is $( 20 \mathbf { i } - 48 \mathbf { j } ) \mathrm { km } \mathrm { h } ^ { - 1 }$ and find the velocity vector of $T$.
Find expressions for the position vectors of $S$ and $T$ at time $t$ minutes after 9.00 a.m.
Show that the bearing of $T$ from $S$ remains constant during the motion, and find this bearing.
Show that if the trains continue at the given speeds they will collide.

Edexcel M1 Q1

\begin{enumerate} \item Two particles, $P$ and $Q$, of mass 2 kg and 1.5 kg respectively are at rest on a smooth, horizontal surface. They are connected by a light, inelastic string which is initially slack. Particle $P$ is projected away from $Q$ with a speed of $7 \mathrm {~ms} ^ { - 1 }$.

Find the common speed of the particles after the string becomes taut.
Calculate the impulse in the string when it jerks tight. \item Particle $A$ has velocity $( 8 \mathbf { i } - 3 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }$ and particle $B$ has velocity $( 15 \mathbf { i } - 8 \mathbf { j } ) \mathrm { ms } ^ { - 1 }$ where $\mathbf { i }$ and $\mathbf { j }$ are perpendicular, horizontal unit vectors.

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