CAIE M1 2002 June — Question 4 7 marks

Exam BoardCAIE
ModuleM1 (Mechanics 1)
Year2002
SessionJune
Marks7
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicPulley systems
TypeParticle motion with kinematics only
DifficultyStandard +0.2 This is a straightforward M1 mechanics question with standard applications of Newton's laws, friction (μ = F/R), constant acceleration kinematics (v = u + at, v² = u² + 2as), and work-energy principles. All parts follow textbook procedures with no novel problem-solving required, making it slightly easier than average for A-level.
Spec3.02d Constant acceleration: SUVAT formulae3.03t Coefficient of friction: F <= mu*R model3.03v Motion on rough surface: including inclined planes
4 A box of mass 4.5 kg is pulled at a constant speed of \(2 \mathrm {~ms} ^ { - 1 }\) along a rough horizontal floor by a horizontal force of magnitude 15 N .
  1. Find the coefficient of friction between the box and the floor. The horizontal pulling force is now removed. Find
  2. the deceleration of the box in the subsequent motion,
  3. the distance travelled by the box from the instant the horizontal force is removed until the box comes to rest.
  4. A cyclist travels in a straight line from \(A\) to \(B\) with constant acceleration \(0.06 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). His speed at \(A\) is \(3 \mathrm {~ms} ^ { - 1 }\) and his speed at \(B\) is \(6 \mathrm {~ms} ^ { - 1 }\). Find
    1. the time taken by the cyclist to travel from \(A\) to \(B\),
    2. the distance \(A B\).
    3. A car leaves \(A\) at the same instant as the cyclist. The car starts from rest and travels in a straight line to \(B\). The car reaches \(B\) at the same instant as the cyclist. At time \(t \mathrm {~s}\) after leaving \(A\) the speed of the car is \(k t ^ { 2 } \mathrm {~ms} ^ { - 1 }\), where \(k\) is a constant. Find
      (a) the value of \(k\),
      (b) the speed of the car at \(B\).
      1. A lorry \(P\) of mass 15000 kg climbs a straight hill of length 800 m at a steady speed. The hill is inclined at \(2 ^ { \circ }\) to the horizontal. For \(P\) 's journey from the bottom of the hill to the top, find
        (a) the gain in gravitational potential energy,
        (b) the work done by the driving force, which has magnitude 7000 N ,
      2. the work done against the force resisting the motion.
      3. A second lorry, \(Q\), also has mass 15000 kg and climbs the same hill as \(P\). The motion of \(Q\) is subject to a constant resisting force of magnitude 900 N , and \(Q\) s speed falls from \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at the bottom of the hill to \(10 \mathrm {~ms} ^ { - 1 }\) at the top. Find the work done by the driving force as \(Q\) climbs from the bottom of the hill to the top. \includegraphics[max width=\textwidth, alt={}, center]{430f1f9a-7a3a-47a0-b742-daf74e68adfd-3_483_231_1537_973} Particles \(A\) and \(B\), of masses 0.15 kg and 0.25 kg respectively, are attached to the ends of a light inextensible string which passes over a smooth fixed pulley. The system is held at rest with the string taut and with \(A\) and \(B\) at the same horizontal level, as shown in the diagram. The system is then released.
        1. Find the downward acceleration of \(B\). After \(2 \mathrm {~s} B\) hits the floor and comes to rest without rebounding. The string becomes slack and \(A\) moves freely under gravity.
        2. Find the time that elapses until the string becomes taut again.
        3. Sketch on a single diagram the velocity-time graphs for both particles, for the period from their release until the instant that \(B\) starts to move upwards.
Question 4:
Part (i):
AnswerMarks Guidance
Answer/WorkingMark Guidance
\(N = 4.5g,\ F = 15\)B1 Allow inequality for M mark
For using \(\mu = F/N\)M1
Coefficient is \(\frac{1}{3}\) or \(0.333\) (\(0.340\) from \(g = 9.8\) or \(9.81\))A1
Part (ii):
AnswerMarks Guidance
Answer/WorkingMark Guidance
For using Newton's 2nd law \([-15 = 4.5a]\)M1 \(4.5a = 15 \Rightarrow a = 10/3\) scores M1 A0 unless \(a\) is said to be deceleration
Deceleration is \(\frac{10}{3}\) ms\(^{-2}\) (or \(3.33\)) or \(a = -\frac{10}{3}\) (or \(-3.33\))A1
Part (iii):
AnswerMarks Guidance
Answer/WorkingMark Guidance
For using \(v^2 = u^2 + 2as\) or \(v = u + at\) and \(s = \frac{u+v}{2}t\); \([0 = 4 + 2(-\frac{10}{3})s]\)M1 \(v = 2,\ u = 0\) and \(a = 10/3\) is OK for M1 even if \(a \neq +10/3\) from (ii); allow A1 as well if \(0.6\) m found
Distance is \(0.6\) mA1 ft Accept \(0.601\) from \(a = -3.33\) for A mark 
# Question 4:

## Part (i):

| Answer/Working | Mark | Guidance |
|---|---|---|
| $N = 4.5g,\ F = 15$ | B1 | Allow inequality for M mark |
| For using $\mu = F/N$ | M1 | |
| Coefficient is $\frac{1}{3}$ or $0.333$ ($0.340$ from $g = 9.8$ or $9.81$) | A1 | |

## Part (ii):

| Answer/Working | Mark | Guidance |
|---|---|---|
| For using Newton's 2nd law $[-15 = 4.5a]$ | M1 | $4.5a = 15 \Rightarrow a = 10/3$ scores M1 A0 unless $a$ is said to be deceleration |
| Deceleration is $\frac{10}{3}$ ms$^{-2}$ (or $3.33$) or $a = -\frac{10}{3}$ (or $-3.33$) | A1 | |

## Part (iii):

| Answer/Working | Mark | Guidance |
|---|---|---|
| For using $v^2 = u^2 + 2as$ or $v = u + at$ and $s = \frac{u+v}{2}t$; $[0 = 4 + 2(-\frac{10}{3})s]$ | M1 | $v = 2,\ u = 0$ and $a = 10/3$ is OK for M1 even if $a \neq +10/3$ from (ii); allow A1 as well if $0.6$ m found |
| Distance is $0.6$ m | A1 ft | Accept $0.601$ from $a = -3.33$ for A mark |

---
4 A box of mass 4.5 kg is pulled at a constant speed of $2 \mathrm {~ms} ^ { - 1 }$ along a rough horizontal floor by a horizontal force of magnitude 15 N .\\
(i) Find the coefficient of friction between the box and the floor.

The horizontal pulling force is now removed. Find\\
(ii) the deceleration of the box in the subsequent motion,\\
(iii) the distance travelled by the box from the instant the horizontal force is removed until the box comes to rest.\\
(i) A cyclist travels in a straight line from $A$ to $B$ with constant acceleration $0.06 \mathrm {~m} \mathrm {~s} ^ { - 2 }$. His speed at $A$ is $3 \mathrm {~ms} ^ { - 1 }$ and his speed at $B$ is $6 \mathrm {~ms} ^ { - 1 }$. Find
\begin{enumerate}[label=(\alph*)]
\item the time taken by the cyclist to travel from $A$ to $B$,
\item the distance $A B$.\\
(ii) A car leaves $A$ at the same instant as the cyclist. The car starts from rest and travels in a straight line to $B$. The car reaches $B$ at the same instant as the cyclist. At time $t \mathrm {~s}$ after leaving $A$ the speed of the car is $k t ^ { 2 } \mathrm {~ms} ^ { - 1 }$, where $k$ is a constant. Find\\
(a) the value of $k$,\\
(b) the speed of the car at $B$.
\begin{enumerate}[label=(\roman*)]
\item A lorry $P$ of mass 15000 kg climbs a straight hill of length 800 m at a steady speed. The hill is inclined at $2 ^ { \circ }$ to the horizontal. For $P$ 's journey from the bottom of the hill to the top, find\\
(a) the gain in gravitational potential energy,\\
(b) the work done by the driving force, which has magnitude 7000 N ,
\item the work done against the force resisting the motion.
\item A second lorry, $Q$, also has mass 15000 kg and climbs the same hill as $P$. The motion of $Q$ is subject to a constant resisting force of magnitude 900 N , and $Q$ s speed falls from $20 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ at the bottom of the hill to $10 \mathrm {~ms} ^ { - 1 }$ at the top. Find the work done by the driving force as $Q$ climbs from the bottom of the hill to the top.\\
\includegraphics[max width=\textwidth, alt={}, center]{430f1f9a-7a3a-47a0-b742-daf74e68adfd-3_483_231_1537_973}

Particles $A$ and $B$, of masses 0.15 kg and 0.25 kg respectively, are attached to the ends of a light inextensible string which passes over a smooth fixed pulley. The system is held at rest with the string taut and with $A$ and $B$ at the same horizontal level, as shown in the diagram. The system is then released.
\begin{enumerate}[label=(\roman*)]
\item Find the downward acceleration of $B$.

After $2 \mathrm {~s} B$ hits the floor and comes to rest without rebounding. The string becomes slack and $A$ moves freely under gravity.
\item Find the time that elapses until the string becomes taut again.
\item Sketch on a single diagram the velocity-time graphs for both particles, for the period from their release until the instant that $B$ starts to move upwards.
\end{enumerate}
\end{enumerate}\end{enumerate}

\hfill \mbox{\textit{CAIE M1 2002 Q4 [7]}}
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