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Cambridge International AS and A Level

Name of student MICHAELERRON Date
Adm. number Year/grade 1989 Stream Michaelerron
Subject Mechanics 1 (M1) Variant(s) P41, P42, P43
Start time Duration Stop time

Qtn No. 1 2 3 4 5 6 Total
Marks 8 6 7 8 8 8 45

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Question 1 Code: 9709/41/M/J/15/5, Topic: -

A cyclist and her bicycle have a total mass of $84 \mathrm{~kg}$. She works at a constant rate of $P \mathrm{~W}$ while moving on a straight road which is inclined to the horizontal at an angle $\theta$, where $\sin \theta=0.1$. When moving uphill, the cyclist's acceleration is $1.25 \mathrm{~m} \mathrm{~s}^{-2}$ at an instant when her speed is $3 \mathrm{~m} \mathrm{~s}^{-1}$. When moving downhill, the cyclist's acceleration is $1.25 \mathrm{~m} \mathrm{~s}^{-2}$ at an instant when her speed is $10 \mathrm{~m} \mathrm{~s}^{-1}$. The resistance to the cyclist's motion, whether the cyclist is moving uphill or downhill, is $R \mathrm{~N}$. Find the values of $P$ and $R$. $[8]$

Question 2 Code: 9709/42/M/J/15/5, Topic: -

A particle $P$ starts from rest at a point $O$ on a horizontal straight line. $P$ moves along the line with constant acceleration and reaches a point $A$ on the line with a speed of $30 \mathrm{~m} \mathrm{~s}^{-1}$. At the instant that $P$ leaves $O$, a particle $Q$ is projected vertically upwards from the point $A$ with a speed of $20 \mathrm{~m} \mathrm{~s}^{-1}$. Subsequently $P$ and $Q$ collide at $A$. Find

$\text{(i)}$ the acceleration of $P$, $[4]$

$\text{(ii)}$ the distance $O A$. $[2]$

Question 3 Code: 9709/43/M/J/15/5, Topic: -


Four coplanar forces of magnitudes $4 \mathrm{~N}, 8 \mathrm{~N}, 12 \mathrm{~N}$ and $16 \mathrm{~N}$ act at a point. The directions in which the forces act are shown in Fig. 1.

$\text{(i)}$ Find the magnitude and direction of the resultant of the four forces. $[5]$


The forces of magnitudes $4 \mathrm{~N}$ and $16 \mathrm{~N}$ exchange their directions and the forces of magnitudes $8 \mathrm{~N}$ and $12 \mathrm{~N}$ also exchange their directions (see Fig. 2).

$\text{(ii)}$ State the magnitude and direction of the resultant of the four forces in Fig. $2.$ $[5]$

Question 4 Code: 9709/41/O/N/15/5, Topic: -


A small bead $Q$ can move freely along a smooth horizontal straight wire $A B$ of length $3 \mathrm{~m}$. Three horizontal forces of magnitudes $F \mathrm{~N}, 10 \mathrm{~N}$ and $20 \mathrm{~N}$ act on the bead in the directions shown in the diagram. The magnitude of the resultant of the three forces is $R \mathrm{~N}$ in the direction shown in the diagram.

$\text{(i)}$ Find the values of $F$ and $R$. $[5]$

$\text{(ii)}$ Initially the bead is at rest at $A$. It reaches $B$ with a speed of $11.7 \mathrm{~m} \mathrm{~s}^{-1}$. Find the mass of the bead. $[3]$

Question 5 Code: 9709/42/O/N/15/5, Topic: -


A smooth inclined plane of length $2.5 \mathrm{~m}$ is fixed with one end on the horizontal floor and the other end at a height of $0.7 \mathrm{~m}$ above the floor. Particles $P$ and $Q$, of masses $0.5 \mathrm{~kg}$ and $0.1 \mathrm{~kg}$ respectively, are attached to the ends of a light inextensible string which passes over a small smooth pulley fixed at the top of the plane. Particle $Q$ is held at rest on the floor vertically below the pulley. The string is taut and $P$ is at rest on the plane (see diagram). $Q$ is released and starts to move vertically upwards towards the pulley and $P$ moves down the plane.

$\text{(i)}$ Find the tension in the string and the magnitude of the acceleration of the particles before $Q$ reaches the pulley. $[5]$

At the instant just before $Q$ reaches the pulley the string breaks; $P$ continues to move down the plane and reaches the floor with a speed of $2 \mathrm{~m} \mathrm{~s}^{-1}$.

$\text{(ii)}$ Find the length of the string. $[3]$

Question 6 Code: 9709/43/O/N/15/5, Topic: -

A cyclist and his bicycle have a total mass of $90 \mathrm{~kg}$. The cyclist starts to move with speed $3 \mathrm{~ms}^{-1}$ from the top of a straight hill, of length $500 \mathrm{~m}$, which is inclined at an angle of $\sin ^{-1} 0.05$ to the horizontal. The cyclist moves with constant acceleration until he reaches the bottom of the hill with speed $5 \mathrm{~m} \mathrm{~s}^{-1}$. The cyclist generates $420 \mathrm{~W}$ of power while moving down the hill. The resistance to the motion of the cyclist and his bicycle, $R \mathrm{~N}$, and the cyclist's speed, $v \mathrm{~m} \mathrm{~s}^{-1}$, both vary.

$\text{(i)}$ Show that $\displaystyle R=\frac{420}{v}+43.56$ $[5]$

$\text{(ii)}$ Find the cyclist's speed at the mid-point of the hill. Hence find the decrease in the value of $R$ when the cyclist moves from the top of the hill to the mid-point of the hill, and when the cyclist moves from the mid-point of the hill to the bottom of the hill. $[3]$

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