OCR M1 — Question 6 12 marks

Exam BoardOCR
ModuleM1 (Mechanics 1)
Marks12
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicTravel graphs
TypeMulti-stage motion with velocity-time graph given
DifficultyStandard +0.3 This is a straightforward kinematics problem using velocity-time graphs. It requires basic application of SUVAT equations and understanding that area under v-t graph equals displacement. All parts involve routine calculations with clearly stated information and no problem-solving insight needed—slightly easier than average A-level.
Spec3.02c Interpret kinematic graphs: gradient and area3.02d Constant acceleration: SUVAT formulae

\includegraphics{figure_6} The diagram shows the \((t, v)\) graph for the motion of a hoist used to deliver materials to different levels at a building site. The hoist moves vertically. The graph consists of straight line segments. In the first stage the hoist travels upwards from ground level for \(25\) s, coming to rest \(8\) m above ground level.
  1. Find the greatest speed reached by the hoist during this stage. [2]
The second stage consists of a \(40\) s wait at the level reached during the first stage. In the third stage the hoist continues upwards until it comes to rest \(40\) m above ground level, arriving \(135\) s after leaving ground level. The hoist accelerates at \(0.02\) m s\(^{-2}\) for the first \(40\) s of the third stage, reaching a speed of \(V\) m s\(^{-1}\). Find
  1. the value of \(V\), [3]
  2. the length of time during the third stage for which the hoist is moving at constant speed, [4]
  3. the deceleration of the hoist in the final part of the third stage. [3]

Question 6:
6
3
June 2011
5
R
Q
P
ThreeparticlesP,QandRlieonalineofgreatestslopeofasmoothinclinedplane. Phasmass0.5kg
andinitiallyisatthefootoftheplane. Rhasmass0.3kgandinitiallyisatthetopoftheplane. Qhas
mass 0.2kg and is between P and R (see diagram). P is projected up the line of greatest slope with
speed3ms −1 attheinstantwhenQandRarereleasedfromrest. Eachparticlehasanaccelerationof
2.5ms
−2
downtheplane.
(i) P and Q collide 0.4s after being set in motion. Immediately after the collision Q moves up the
plane with speed 3.2ms −1. Find the speed and direction of motion of P immediately after the
collision. [5]
(ii) 0.6s after its collision with P, Q collides with R and the two particles coalesce. Find the speed
anddirectionofmotionof thecombinedparticleimmediatelyafter thecollision [5]
6
A B
q
5 N
R
7 N
AsmallsmoothringRofweight7Nisthreadedonalightinextensiblestring. Theendsofthestring
areattachedtofixedpointsAandBatthesamehorizontallevel. Ahorizontalforceofmagnitude5N
isappliedtoR. Thestringistaut. Inthe equilibriumpositiontheangleARBisa rightangle, andthe
portionofthestringattachedtoBmakesanangleθwiththe horizontal(seediagram).
(i) ExplainwhythetensionTN isthesameineachpartofthestring. [1]
(ii) By resolving horizontally and vertically for the forces acting on R, form two simultaneous
equationsinTcosθandTsinθ. [4]
(iii) Hence findT andθ. [6]
[Question7isprintedoverleaf.]
Turnover
©OCR2011 4728 Jun11
4
June 2011
7 A particle P is projected from a fixed point O on a straight line. The displacement xm of P from O
attimetsafterprojectionisgivenbyx =0.1t3−0.3t2+0.2t.
(i) ExpressthevelocityandaccelerationofPintermsoft. [4]
(ii) ShowthatwhentheaccelerationofPiszero,PisatO. [3]
(iii) Findthevaluesoft whenPisstationary. [3]
At the instantwhenP firstleavesO, a particle Q is projected fromO. Q moves onthe same straight
line as P and at time ts after projection the velocity of Q is given by (0.2t2 −0.4)ms −1. P and Q
collidefirstwhent = T.
(iv) ShowthatT satisfiestheequationt2−9t+18=0,andhencefindT. [7]
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AcknowledgementsBooklet.Thisisproducedforeachseriesofexaminationsandisfreelyavailabletodownloadfromourpublicwebsite(www.ocr.org.uk)aftertheliveexaminationseries.
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oftheUniversityofCambridge.
©OCR2011 4728 Jun11
2
Jan 2012
1 Particles P and Q, of masses 0.3 kg and 0.5 kg respectively, are moving in the same direction along the same
straight line on a smooth horizontal surface. P is moving with speed 2.2 m s−1 and Q is moving with speed
0.8 m s−1 immediately before they collide. In the collision, the speed of P is reduced by 50% and its direction
of motion is unchanged.
(i) Calculate the speed of Q immediately after the collision. [4]
(ii) Find the distance PQ at the instant 3 seconds after the collision. [2]
2 In the sport of curling, a heavy stone is projected across a horizontal ice surface. One player projects a stone
of weight 180 N, which moves 36 m in a straight line and comes to rest 24 s after the instant of projection.
The only horizontal force acting on the stone after its projection is a constant frictional force between the
stone and the ice.
(i) Calculate the deceleration of the stone. [2]
(ii) Find the magnitude of the frictional force acting on the stone, and calculate the coefficient of friction
between the stone and the ice. [4]
3 A car is travelling along a straight horizontal road with velocity 32.5 m s−1. The driver applies the brakes
and the car decelerates at (8 − 0.6t) m s−2, where t s is the time which has elapsed since the brakes were first
applied.
(i) Show that, while the car is decelerating, its velocity is (32.5 − 8t + 0.3t2) m s−1. [3]
(ii) Find the time taken to bring the car to rest. [2]
(iii) Show that the distance travelled while the car is decelerating is 75 m. [4]
4
8 N
150° 15 N
120°
20 N
Three horizontal forces of magnitudes 8 N, 15 N and 20 N act at a point. The 8 N and 15 N forces are at right
angles. The 20 N force makes an angle of 150° with the 8 N force and an angle of 120° with the 15 N force
(see diagram).
(i) Calculate the components of the resultant force in the directions of the 8 N and 15 N forces. [3]
(ii) Calculate the magnitude of the resultant force, and the angle it makes with the direction of the 8 N
force. [4]
The directions in which the three horizontal forces act can be altered.
(iii) State the greatest and least possible magnitudes of the resultant force. [2]
© OCR 2012 4728 Jan12
Jan 2012 4
7
Q
R
P
Particles P and Q, of masses m kg and 0.05 kg respectively, are attached to the ends of a light inextensible
string which passes over a smooth pulley. Q is attached to a particle R of mass 0.45 kg by a light inextensible
string. The strings are taut, and the portions of the strings not in contact with the pulley are vertical. P is in
contact with a horizontal surface when the particles are released from rest (see diagram). The tension in the
string QR is 2.52 N during the descent of R.
(i) (a) Find the acceleration of R during its descent. [2]
(b) By considering the motion of Q, calculate the tension in the string PQ during the descent of R. [3]
(ii) Find the value of m. [3]
R strikes the surface 0.5 s after release and does not rebound. During their subsequent motion, P does not
reach the pulley and Q does not reach the surface.
(iii) Calculate the greatest height of P above the surface. [8]
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© OCR 2012 4728 Jan12
Jan 2012
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June 2014 2
3.5ms-1
1 A particle P is projected vertically downwards with initial speed from a point A which is 5 m
above horizontal ground.
(i) Find the speed of P immediately before it strikes the ground. [2]
After striking the ground, P rebounds and moves vertically upwards and 0.87 s after leaving the ground P
passes through A.
(ii) Calculate the speed of P immediately after it leaves the ground. [3]
It is given that the mass of P is 0.2 kg.
(iii) Calculate the change in the momentum of P as a result of its collision with the ground. [2]
2
2.5 N
2.4 N i°
F N
A particle rests on a smooth horizontal surface. Three horizontal forces of magnitudes 2.5 N, F N and 2.4 N
act on the particle on bearings i°, 180° and 270° respectively (see diagram). The particle is in equilibrium.
(i) Find i and F. [4]
The 2.4 N force suddenly ceases to act on the particle, which has mass 0.2 kg.
(ii) Find the magnitude and direction of the acceleration of the particle. [3]
3 A particle P travels in a straight line. The velocity of P at time t seconds after it passes through a fixed point
A is given by
(0.6
t
2 +3)ms-1
. Find
(i) the velocity of P when it passes through A, [1]
(ii) the displacement of P from A when t
=1.5
, [4]
6ms-2
(iii) the velocity of P when it has acceleration . [3]
© OCR 2014 4728/01 Jun14
3
June 2014
4
4 m s–1 2 m s–1
P Q
4ms-1 2ms-1
Particles P and Q are moving towards each other with constant speeds and along the same
straight line on a smooth horizontal surface (see diagram). P has mass 0.2 kg and Q has mass 0.3 kg. The
two particles collide.
(i) Show that Q must change its direction of motion in the collision. [3]
(ii) Given that P and Q move with equal speed after the collision, calculate both possible values for their
speed after they collide. [5]
5
x (m)
5
O
T 3 5 7 9 10
t (s)
A particle P can move in a straight line on a horizontal surface. At time t seconds the displacement of P from
a fixed point A on the line is x m. The diagram shows the (t, x) graph for P. In the interval 0G t G 10 , either
4ms-1
the speed of P is , or P is at rest.
(i) Show by calculation that T
=1.75
. [2]
(ii) State the velocity of P when
(a) t =2, [1]
(b) t =8, [1]
(c) t =9. [1]
(iii) Calculate the distance travelled by P in the interval 0G t G 10 . [3]
For t 210 , the displacement of P from A is given by x=20 t-t 2 -96 .
(iv) Calculate the value of t, where t
210
, for which the speed of P is
4ms-1
. [4]
© OCR 2014 4728/01 Jun14 Turn over
4
June 2014
6 A particle P of weight 8 N rests on a horizontal surface. A horizontal force of magnitude 3 N acts on P, and
P is in limiting equilibrium.
(i) Calculate the coefficient of friction between P and the surface. [2]
(ii) Find the magnitude and direction of the contact force exerted by the surface on P. [4]
(iii)
T N
P
i° 3 N
The initial 3 N force continues to act on P in its original direction. An additional force of magnitude
T N, acting in the same vertical plane as the 3 N force, is now applied to P at an angle of i° above the
horizontal (see diagram). P is again in limiting equilibrium.
(a) Given that i=0, find T. [2]
(b) Given instead that
i=30
, calculate T. [6]
© OCR 2014 4728/01 Jun14
5
June 2014
7
0.6 m s–1
P
A
0.6 m s–1
Q
M
30°
B
A and B are points at the upper and lower ends, respectively, of a line of greatest slope on a plane inclined
at 30° to the horizontal. M is the mid-point of AB. Two particles P and Q, joined by a taut light inextensible
string, are placed on the plane at A and M respectively. The particles are simultaneously projected with
0.6ms-1
speed down the line of greatest slope (see diagram). The particles move down the plane with
0.9ms-2
acceleration . At the instant 2 s after projection, P is at M and Q is at B. The particle Q subsequently
remains at rest at B.
(i) Find the distance AB. [3]
The plane is rough between A and M, but smooth between M and B.
(ii) Calculate the speed of P when it reaches B. [4]
P has mass 0.4 kg and Q has mass 0.3 kg.
(iii) By considering the motion of Q, calculate the tension in the string while both particles are moving
down the plane. [3]
(iv) Calculate the coefficient of friction between P and the plane between A and M. [6]
END OF QUESTION PAPER
© OCR 2014 4728/01 Jun14
Question 6:
6
3
June 2011
5
R
Q
P
ThreeparticlesP,QandRlieonalineofgreatestslopeofasmoothinclinedplane. Phasmass0.5kg
andinitiallyisatthefootoftheplane. Rhasmass0.3kgandinitiallyisatthetopoftheplane. Qhas
mass 0.2kg and is between P and R (see diagram). P is projected up the line of greatest slope with
speed3ms −1 attheinstantwhenQandRarereleasedfromrest. Eachparticlehasanaccelerationof
2.5ms
−2
downtheplane.
(i) P and Q collide 0.4s after being set in motion. Immediately after the collision Q moves up the
plane with speed 3.2ms −1. Find the speed and direction of motion of P immediately after the
collision. [5]
(ii) 0.6s after its collision with P, Q collides with R and the two particles coalesce. Find the speed
anddirectionofmotionof thecombinedparticleimmediatelyafter thecollision [5]
6
A B
q
5 N
R
7 N
AsmallsmoothringRofweight7Nisthreadedonalightinextensiblestring. Theendsofthestring
areattachedtofixedpointsAandBatthesamehorizontallevel. Ahorizontalforceofmagnitude5N
isappliedtoR. Thestringistaut. Inthe equilibriumpositiontheangleARBisa rightangle, andthe
portionofthestringattachedtoBmakesanangleθwiththe horizontal(seediagram).
(i) ExplainwhythetensionTN isthesameineachpartofthestring. [1]
(ii) By resolving horizontally and vertically for the forces acting on R, form two simultaneous
equationsinTcosθandTsinθ. [4]
(iii) Hence findT andθ. [6]
[Question7isprintedoverleaf.]
Turnover
©OCR2011 4728 Jun11
4
June 2011
7 A particle P is projected from a fixed point O on a straight line. The displacement xm of P from O
attimetsafterprojectionisgivenbyx =0.1t3−0.3t2+0.2t.
(i) ExpressthevelocityandaccelerationofPintermsoft. [4]
(ii) ShowthatwhentheaccelerationofPiszero,PisatO. [3]
(iii) Findthevaluesoft whenPisstationary. [3]
At the instantwhenP firstleavesO, a particle Q is projected fromO. Q moves onthe same straight
line as P and at time ts after projection the velocity of Q is given by (0.2t2 −0.4)ms −1. P and Q
collidefirstwhent = T.
(iv) ShowthatT satisfiestheequationt2−9t+18=0,andhencefindT. [7]
CopyrightInformation
OCRiscommittedtoseekingpermissiontoreproduceallthird-partycontentthatitusesinitsassessmentmaterials. OCRhasattemptedtoidentifyandcontactallcopyrightholders
whoseworkisusedinthispaper. Toavoidtheissueofdisclosureofanswer-relatedinformationtocandidates,allcopyrightacknowledgementsarereproducedintheOCRCopyright
AcknowledgementsBooklet.Thisisproducedforeachseriesofexaminationsandisfreelyavailabletodownloadfromourpublicwebsite(www.ocr.org.uk)aftertheliveexaminationseries.
IfOCRhasunwittinglyfailedtocorrectlyacknowledgeorclearanythird-partycontentinthisassessmentmaterial,OCRwillbehappytocorrectitsmistakeattheearliestpossibleopportunity.
ForqueriesorfurtherinformationpleasecontacttheCopyrightTeam,FirstFloor,9HillsRoad,CambridgeCB21GE.
OCRispartoftheCambridgeAssessmentGroup;CambridgeAssessmentisthebrandnameofUniversityofCambridgeLocalExaminationsSyndicate(UCLES),whichisitselfadepartment
oftheUniversityofCambridge.
©OCR2011 4728 Jun11
2
Jan 2012
1 Particles P and Q, of masses 0.3 kg and 0.5 kg respectively, are moving in the same direction along the same
straight line on a smooth horizontal surface. P is moving with speed 2.2 m s−1 and Q is moving with speed
0.8 m s−1 immediately before they collide. In the collision, the speed of P is reduced by 50% and its direction
of motion is unchanged.
(i) Calculate the speed of Q immediately after the collision. [4]
(ii) Find the distance PQ at the instant 3 seconds after the collision. [2]
2 In the sport of curling, a heavy stone is projected across a horizontal ice surface. One player projects a stone
of weight 180 N, which moves 36 m in a straight line and comes to rest 24 s after the instant of projection.
The only horizontal force acting on the stone after its projection is a constant frictional force between the
stone and the ice.
(i) Calculate the deceleration of the stone. [2]
(ii) Find the magnitude of the frictional force acting on the stone, and calculate the coefficient of friction
between the stone and the ice. [4]
3 A car is travelling along a straight horizontal road with velocity 32.5 m s−1. The driver applies the brakes
and the car decelerates at (8 − 0.6t) m s−2, where t s is the time which has elapsed since the brakes were first
applied.
(i) Show that, while the car is decelerating, its velocity is (32.5 − 8t + 0.3t2) m s−1. [3]
(ii) Find the time taken to bring the car to rest. [2]
(iii) Show that the distance travelled while the car is decelerating is 75 m. [4]
4
8 N
150° 15 N
120°
20 N
Three horizontal forces of magnitudes 8 N, 15 N and 20 N act at a point. The 8 N and 15 N forces are at right
angles. The 20 N force makes an angle of 150° with the 8 N force and an angle of 120° with the 15 N force
(see diagram).
(i) Calculate the components of the resultant force in the directions of the 8 N and 15 N forces. [3]
(ii) Calculate the magnitude of the resultant force, and the angle it makes with the direction of the 8 N
force. [4]
The directions in which the three horizontal forces act can be altered.
(iii) State the greatest and least possible magnitudes of the resultant force. [2]
© OCR 2012 4728 Jan12
Jan 2012 4
7
Q
R
P
Particles P and Q, of masses m kg and 0.05 kg respectively, are attached to the ends of a light inextensible
string which passes over a smooth pulley. Q is attached to a particle R of mass 0.45 kg by a light inextensible
string. The strings are taut, and the portions of the strings not in contact with the pulley are vertical. P is in
contact with a horizontal surface when the particles are released from rest (see diagram). The tension in the
string QR is 2.52 N during the descent of R.
(i) (a) Find the acceleration of R during its descent. [2]
(b) By considering the motion of Q, calculate the tension in the string PQ during the descent of R. [3]
(ii) Find the value of m. [3]
R strikes the surface 0.5 s after release and does not rebound. During their subsequent motion, P does not
reach the pulley and Q does not reach the surface.
(iii) Calculate the greatest height of P above the surface. [8]
Copyright Information
OCR is committed to seeking permission to reproduce all third-party content that it uses in its assessment materials. OCR has attempted to identify and contact all copyright holders
whose work is used in this paper. To avoid the issue of disclosure of answer-related information to candidates, all copyright acknowledgements are reproduced in the OCR Copyright
Acknowledgements Booklet. This is produced for each series of examinations and is freely available to download from our public website (www.ocr.org.uk) after the live examination series.
If OCR has unwittingly failed to correctly acknowledge or clear any third-party content in this assessment material, OCR will be happy to correct its mistake at the earliest possible
opportunity.
For queries or further information please contact the Copyright Team, First Floor, 9 Hills Road, Cambridge CB2 1GE.
OCR is part of the Cambridge Assessment Group; Cambridge Assessment is the brand name of University of Cambridge Local Examinations Syndicate (UCLES), which is itself a
department of the University of Cambridge.
© OCR 2012 4728 Jan12
Jan 2012
June 2012 | 2
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JaJnu n2e0 122012 | 3
une 2012
Jan 2012 | 4
JJaann 2 2001123
JJaann 2 2001123 | 2
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2
JuJnaen 22001132
JuJnaen 22001132
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June 2013
Jan 2012
June 2013 | 3
Jan 2012
June 2013 | 4
June 2014 2
3.5ms-1
1 A particle P is projected vertically downwards with initial speed from a point A which is 5 m
above horizontal ground.
(i) Find the speed of P immediately before it strikes the ground. [2]
After striking the ground, P rebounds and moves vertically upwards and 0.87 s after leaving the ground P
passes through A.
(ii) Calculate the speed of P immediately after it leaves the ground. [3]
It is given that the mass of P is 0.2 kg.
(iii) Calculate the change in the momentum of P as a result of its collision with the ground. [2]
2
2.5 N
2.4 N i°
F N
A particle rests on a smooth horizontal surface. Three horizontal forces of magnitudes 2.5 N, F N and 2.4 N
act on the particle on bearings i°, 180° and 270° respectively (see diagram). The particle is in equilibrium.
(i) Find i and F. [4]
The 2.4 N force suddenly ceases to act on the particle, which has mass 0.2 kg.
(ii) Find the magnitude and direction of the acceleration of the particle. [3]
3 A particle P travels in a straight line. The velocity of P at time t seconds after it passes through a fixed point
A is given by
(0.6
t
2 +3)ms-1
. Find
(i) the velocity of P when it passes through A, [1]
(ii) the displacement of P from A when t
=1.5
, [4]
6ms-2
(iii) the velocity of P when it has acceleration . [3]
© OCR 2014 4728/01 Jun14
3
June 2014
4
4 m s–1 2 m s–1
P Q
4ms-1 2ms-1
Particles P and Q are moving towards each other with constant speeds and along the same
straight line on a smooth horizontal surface (see diagram). P has mass 0.2 kg and Q has mass 0.3 kg. The
two particles collide.
(i) Show that Q must change its direction of motion in the collision. [3]
(ii) Given that P and Q move with equal speed after the collision, calculate both possible values for their
speed after they collide. [5]
5
x (m)
5
O
T 3 5 7 9 10
t (s)
A particle P can move in a straight line on a horizontal surface. At time t seconds the displacement of P from
a fixed point A on the line is x m. The diagram shows the (t, x) graph for P. In the interval 0G t G 10 , either
4ms-1
the speed of P is , or P is at rest.
(i) Show by calculation that T
=1.75
. [2]
(ii) State the velocity of P when
(a) t =2, [1]
(b) t =8, [1]
(c) t =9. [1]
(iii) Calculate the distance travelled by P in the interval 0G t G 10 . [3]
For t 210 , the displacement of P from A is given by x=20 t-t 2 -96 .
(iv) Calculate the value of t, where t
210
, for which the speed of P is
4ms-1
. [4]
© OCR 2014 4728/01 Jun14 Turn over
4
June 2014
6 A particle P of weight 8 N rests on a horizontal surface. A horizontal force of magnitude 3 N acts on P, and
P is in limiting equilibrium.
(i) Calculate the coefficient of friction between P and the surface. [2]
(ii) Find the magnitude and direction of the contact force exerted by the surface on P. [4]
(iii)
T N
P
i° 3 N
The initial 3 N force continues to act on P in its original direction. An additional force of magnitude
T N, acting in the same vertical plane as the 3 N force, is now applied to P at an angle of i° above the
horizontal (see diagram). P is again in limiting equilibrium.
(a) Given that i=0, find T. [2]
(b) Given instead that
i=30
, calculate T. [6]
© OCR 2014 4728/01 Jun14
5
June 2014
7
0.6 m s–1
P
A
0.6 m s–1
Q
M
30°
B
A and B are points at the upper and lower ends, respectively, of a line of greatest slope on a plane inclined
at 30° to the horizontal. M is the mid-point of AB. Two particles P and Q, joined by a taut light inextensible
string, are placed on the plane at A and M respectively. The particles are simultaneously projected with
0.6ms-1
speed down the line of greatest slope (see diagram). The particles move down the plane with
0.9ms-2
acceleration . At the instant 2 s after projection, P is at M and Q is at B. The particle Q subsequently
remains at rest at B.
(i) Find the distance AB. [3]
The plane is rough between A and M, but smooth between M and B.
(ii) Calculate the speed of P when it reaches B. [4]
P has mass 0.4 kg and Q has mass 0.3 kg.
(iii) By considering the motion of Q, calculate the tension in the string while both particles are moving
down the plane. [3]
(iv) Calculate the coefficient of friction between P and the plane between A and M. [6]
END OF QUESTION PAPER
© OCR 2014 4728/01 Jun14
\includegraphics{figure_6}

The diagram shows the $(t, v)$ graph for the motion of a hoist used to deliver materials to different levels at a building site. The hoist moves vertically. The graph consists of straight line segments. In the first stage the hoist travels upwards from ground level for $25$ s, coming to rest $8$ m above ground level.

\begin{enumerate}[label=(\roman*)]
\item Find the greatest speed reached by the hoist during this stage. [2]
\end{enumerate}

The second stage consists of a $40$ s wait at the level reached during the first stage. In the third stage the hoist continues upwards until it comes to rest $40$ m above ground level, arriving $135$ s after leaving ground level. The hoist accelerates at $0.02$ m s$^{-2}$ for the first $40$ s of the third stage, reaching a speed of $V$ m s$^{-1}$. Find

\begin{enumerate}[label=(\roman*)]
\setcounter{enumi}{1}
\item the value of $V$, [3]
\item the length of time during the third stage for which the hoist is moving at constant speed, [4]
\item the deceleration of the hoist in the final part of the third stage. [3]
\end{enumerate}

\hfill \mbox{\textit{OCR M1  Q6 [12]}}