$\require{\cancel}$ $\require{\stix[upint]}$

Qtn No.	1	2	3	4	5	6	Total
Marks	11	10	10	10	11	10	62
Score

Question 1 Code: 9709/41/M/J/11/7, Topic: -

Loads $A$ and $B$, of masses $1.2 \mathrm{~kg}$ and $2.0 \mathrm{~kg}$ respectively, are attached to the ends of a light inextensible string which passes over a fixed smooth pulley. $A$ is held at rest and $B$ hangs freely, with both straight parts of the string vertical. $A$ is released and starts to move upwards. It does not reach the pulley in the subsequent motion.

$\text{(i)}$ Find the acceleration of $A$ and the tension in the string. $[4]$

$\text{(ii)}$ Find, for the first $1.5$ metres of $A$ 's motion,

$\text{(a)}$ A's gain in potential energy,

$\text{(b)}$ the work done on $A$ by the tension in the string,

$\text{(c)}$ A's gain in kinetic energy.

$[3]$

$B$ hits the floor $1.6$ seconds after $A$ is released. $B$ comes to rest without rebounding and the string becomes slack.

$\text{(iii)}$ Find the time from the instant the string becomes slack until it becomes taut again. $[4]$

Question 2 Code: 9709/42/M/J/11/7, Topic: -

A walker travels along a straight road passing through the points $A$ and $B$ on the road with speeds $0.9 \mathrm{~m} \mathrm{~s}^{-1}$ and $1.3 \mathrm{~m} \mathrm{~s}^{-1}$ respectively. The walker's acceleration between $A$ and $B$ is constant and equal to $0.004 \mathrm{~m} \mathrm{~s}^{-2}$.

$\text{(i)}$ Find the time taken by the walker to travel from $A$ to $B$, and find the distance $A B$. $[3]$

A cyclist leaves $A$ at the same instant as the walker. She starts from rest and travels along the straight road, passing through $B$ at the same instant as the walker. At time $t \mathrm{~s}$ after leaving $A$ the cyclist's speed is $k t^{3} \mathrm{~m} \mathrm{~s}^{-1}$, where $k$ is a constant.

$\text{(ii)}$ Show that when $t=64.05$ the speed of the walker and the speed of the cyclist are the same, correct to 3 significant figures. $[5]$

$\text{(ii)}$ Find the cyclist's acceleration at the instant she passes through $B$. $[2]$

Question 3 Code: 9709/43/M/J/11/7, Topic: -

A particle travels in a straight line from $A$ to $B$ in $20 \mathrm{~s}$. Its acceleration $t$ seconds after leaving $A$ is $a \mathrm{~m} \mathrm{~s}^{-2}$, where $\displaystyle a=\frac{3}{160} t^{2}-\frac{1}{800} t^{3}$. It is given that the particle comes to rest at $B$.

$\text{(i)}$ Show that the initial speed of the particle is zero. $[4]$

$\text{(ii)}$ Find the maximum speed of the particle. $[2]$

$\text{(iii)}$ Find the distance $A B$. $[4]$

Question 4 Code: 9709/41/O/N/11/7, Topic: -

A particle $P$ starts from a point $O$ and moves along a straight line. $P$ 's velocity $t$ s after leaving $O$ is $v \mathrm{~m} \mathrm{~s}^{-1}$, where

$$ v=0.16 t^{\frac{3}{2}}-0.016 t^{2} $$

$P$ comes to rest instantaneously at the point $A$.

$\text{(i)}$ Verify that the value of $t$ when $P$ is at $A$ is 100. $[1]$

$\text{(ii)}$ Find the maximum speed of $P$ in the interval $0< t <100$. $[4]$

$\text{(iii)}$ Find the distance $O A$. $[3]$

$\text{(iv)}$ Find the value of $t$ when $P$ passes through $O$ on returning from $A$. $[2]$

Question 5 Code: 9709/42/O/N/11/7, Topic: -

A tractor travels in a straight line from a point $A$ to a point $B$. The velocity of the tractor is $v \mathrm{~m} \mathrm{~s}^{-1}$ at time $t \mathrm{~s}$ after leaving $A$.

$\text{(i)}$

The diagram shows an approximate velocity-time graph for the motion of the tractor. The graph consists of two straight line segments. Use the graph to find an approximation for

$\text{(a)}$ the distance $A B$, $[2]$

$\text{(b)}$ the acceleration of the tractor for $0< t <400$ and for $400< t <800$. $[2]$

$\text{(ii)}$ The actual velocity of the tractor is given by $v=0.04 t-0.00005 t^{2}$ for $0 \leqslant t \leqslant 800$.

$\text{(a)}$ Find the values of $t$ for which the actual acceleration of the tractor is given correctly by the approximate velocity-time graph in part $\text{(i)}$. $[3]$

For the interval $0 \leqslant t \leqslant 400$, the approximate velocity of the tractor in part $\text{(i)}$ is denoted by $v_{1} \mathrm{~m} \mathrm{~s}^{-1}$

$\text{(b)}$ Express $v_{1}$ in terms of $t$ and hence show that $v_{1}-v=0.00005(t-200)^{2}-1$. $[2]$

$\text{(c)}$ Deduce that $-1 \leqslant v_{1}-v \leqslant 1$. $[2]$

Question 6 Code: 9709/43/O/N/11/7, Topic: -

A car of mass $600 \mathrm{~kg}$ travels along a straight horizontal road starting from a point $A$. The resistance to motion of the car is $750 \mathrm{~N}$.

$\text{(i)}$ The car travels from $A$ to $B$ at constant speed in $100 \mathrm{~s}$. The power supplied by the car's engine is constant and equal to $30 \mathrm{~kW}$. Find the distance $A B$. $[3]$

$\text{(ii)}$ The car's engine is switched off at $B$ and the car's speed decreases until the car reaches $C$ with a speed of $20 \mathrm{~m} \mathrm{~s}^{-1}$. Find the distance $B C$. $[3]$

$\text{(iii)}$ The car's engine is switched on at $C$ and the power it supplies is constant and equal to $30 \mathrm{~kW}$. The car takes $14 \mathrm{~s}$ to travel from $C$ to $D$ and reaches $D$ with a speed of $30 \mathrm{~m} \mathrm{~s}^{-1}$. Find the distance $C D.$ $[4]$