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Cambridge International AS and A Level

Name of student BOBBYGROOR Date
Adm. number Year/grade 1995 Stream Bobbygroor
Subject Mechanics 1 (M1) Variant(s) P41, P42, P43
Start time Duration Stop time

Qtn No. 1 2 3 4 5 6 Total
Marks 10 11 9 10 10 9 59

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Question 1 Code: 9709/41/M/J/15/6, Topic: -

Two particles $A$ and $B$ start to move at the same instant from a point $O$. The particles move in the same direction along the same straight line. The acceleration of $A$ at time $t \mathrm{~s}$ after starting to move is $a \mathrm{~m} \mathrm{~s}^{-2}$, where $a=0.05-0.0002 t$.

$\text{(i)}$ Find $A$ 's velocity when $t=200$ and when $t=500$. $[4]$

$B$ moves with constant acceleration for the first $200 \mathrm{~s}$ and has the same velocity as $A$ when $t=200. B$ moves with constant retardation from $t=200$ to $t=500$ and has the same velocity as $A$ when $t=500$.

$\text{(ii)}$ Find the distance between $A$ and $B$ when $t=500$. $[6]$

Question 2 Code: 9709/42/M/J/15/6, Topic: -


Two particles $P$ and $Q$ have masses $m \mathrm{~kg}$ and $(1-m) \mathrm{kg}$ respectively. The particles are attached to the ends of a light inextensible string which passes over a smooth fixed pulley. $P$ is held at rest with the string taut and both straight parts of the string vertical. $P$ and $Q$ are each at a height of $h \mathrm{~m}$ above horizontal ground (see Fig. 1). $P$ is released and $Q$ moves downwards. Subsequently $Q$ hits the ground and comes to rest. Fig. 2 shows the velocity-time graph for $P$ while $Q$ is moving downwards or is at rest on the ground.

$\text{(i)}$ Find the value of $\mathrm{h}$. $[2]$

$\text{(ii)}$ Find the value of $m$, and find also the tension in the string while $Q$ is moving. $[6]$

$\text{(iii)}$ The string is slack while $Q$ is at rest on the ground. Find the total time from the instant that $P$ is released until the string becomes taut again. $[3]$

Question 3 Code: 9709/43/M/J/15/6, Topic: -

A small box of mass $5 \mathrm{~kg}$ is pulled at a constant speed of $2.5 \mathrm{~m} \mathrm{~s}^{-1}$ down a line of greatest slope of a rough plane inclined at $10^{\circ}$ to the horizontal. The pulling force has magnitude $20 \mathrm{~N}$ and acts downwards parallel to a line of greatest slope of the plane.

$\text{(i)}$ Find the coefficient of friction between the box and the plane. $[5]$

The pulling force is removed while the box is moving at $2.5 \mathrm{~m} \mathrm{~s}^{-1}$.

$\text{(ii)}$ Find the distance moved by the box after the instant at which the pulling force is removed. []

Question 4 Code: 9709/41/O/N/15/6, Topic: -

A particle $P$ moves in a straight line, starting from a point $O$. The velocity of $P$, measured in $\mathrm{m} \mathrm{s}^{-1}$, at time $t \mathrm{~s}$ after leaving $O$ is given by

$$ v=0.6 t-0.03 t^{2} $$

$\text{(i)}$ Verify that, when $t=5$, the particle is $6.25 \mathrm{~m}$ from $O$. Find the acceleration of the particle at this time. $[4]$

$\text{(ii)}$ Find the values of $t$ at which the particle is travelling at half of its maximum velocity. $[6]$

Question 5 Code: 9709/42/O/N/15/6, Topic: -


A small ring of mass $0.024 \mathrm{~kg}$ is threaded on a fixed rough horizontal rod. A light inextensible string is attached to the ring and the string is pulled with a force of magnitude $0.195 \mathrm{~N}$ at an angle of $\theta$ with the horizontal, where $\sin \theta=\frac{5}{13}$. When the angle $\theta$ is below the horizontal (see Fig. 1) the ring is in limiting equilibrium.

$\text{(i)}$ Find the coefficient of friction between the ring and the rod. $[6]$

When the angle $\theta$ is above the horizontal (see Fig. 2) the ring moves.

$\text{(ii)}$ Find the acceleration of the ring. $[4]$

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

A particle $P$ starts from rest at a point $O$ of a straight line and moves along the line. The displacement of the particle at time $t \mathrm{~s}$ after leaving $O$ is $x \mathrm{~m}$, where

$$ x=0.08 t^{2}-0.0002 t^{3} $$

$\text{(i)}$ Find the value of $t$ when $P$ returns to $O$ and find the speed of $P$ as it passes through $O$ on its return. $[4]$

$\text{(ii)}$ For the motion of $P$ until the instant it returns to $O$, find

$\text{(a)}$ the total distance travelled, $[3]$

$\text{(b)}$ the average speed. $[2]$

Worked solutions: P1, P3 & P6 (S1)

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