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HENRYTAIGO

Cambridge International AS and A Level

Name of student HENRYTAIGO Date
Adm. number Year/grade HenryTaigo Stream HenryTaigo
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 7 8 7 9 8 47
Score

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Question 1 Code: 9709/41/M/J/13/5, Topic: -

 

A light inextensible string has a particle $A$ of mass $0.26 \mathrm{~kg}$ attached to one end and a particle $B$ of mass $0.54 \mathrm{~kg}$ attached to the other end. The particle $A$ is held at rest on a rough plane inclined at angle $\alpha$ to the horizontal, where $\sin \alpha=\frac{5}{13}$. The string is taut and parallel to a line of greatest slope of the plane. The string passes over a small smooth pulley at the top of the plane. Particle $B$ hangs at rest vertically below the pulley (see diagram). The coefficient of friction between $A$ and the plane is $0.2$. Particle $A$ is released and the particles start to move.

$\text{(i)}$ Find the magnitude of the acceleration of the particles and the tension in the string. $[6]$

Particle $A$ reaches the pulley $0.4 \mathrm{~s}$ after starting to move.

$\text{(ii)}$ Find the distance moved by each of the particles. $[2]$

Question 2 Code: 9709/42/M/J/13/5, Topic: -

A car of mass $1000 \mathrm{~kg}$ is travelling on a straight horizontal road. The power of its engine is constant and equal to $P \mathrm{~kW}$. The resistance to motion of the car is $600 \mathrm{~N}$. At an instant when the car's speed is $25 \mathrm{~m} \mathrm{~s}^{-1}$, its acceleration is $0.2 \mathrm{~m} \mathrm{~s}^{-2}$. Find

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

$\text{(ii)}$ the steady speed at which the car can travel. $[3]$

Question 3 Code: 9709/43/M/J/13/5, Topic: -

A particle $P$ is projected vertically upwards from a point on the ground with speed $17 \mathrm{~m} \mathrm{~s}^{-1}$. Another particle $Q$ is projected vertically upwards from the same point with speed $7 \mathrm{~ms}^{-1}$. Particle $Q$ is projected $T$ seconds later than particle $P$.

$\text{(i)}$ Given that the particles reach the ground at the same instant, find the value of $T$. $[2]$

$\text{(ii)}$ At a certain instant when both $P$ and $Q$ are in motion, $P$ is $5 \mathrm{~m}$ higher than $Q.$ Find the magnitude and direction of the velocity of each of the particles at this instant. $[6]$

Question 4 Code: 9709/41/O/N/13/5, Topic: -

A lorry of mass $15000 \mathrm{~kg}$ climbs from the bottom to the top of a straight hill, of length $1440 \mathrm{~m}$, at a constant speed of $15 \mathrm{~m} \mathrm{~s}^{-1}$. The top of the hill is $16 \mathrm{~m}$ above the level of the bottom of the hill. The resistance to motion is constant and equal to $1800 \mathrm{~N}$.

$\text{(i)}$ Find the work done by the driving force. $[4]$

On reaching the top of the hill the lorry continues on a straight horizontal road and passes through a point $P$ with speed $24 \mathrm{~m} \mathrm{~s}^{-1}$. The resistance to motion is constant and is now equal to $1600 \mathrm{~N}$. The work done by the lorry's engine from the top of the hill to the point $P$ is $5030 \mathrm{~kJ}$.

$\text{(ii)}$ Find the distance from the top of the hill to the point $P$. $[3]$

Question 5 Code: 9709/42/O/N/13/5, Topic: -

A particle $P$ moves in a straight line. $P$ starts from rest at $O$ and travels to $A$ where it comes to rest, taking 50 seconds. The speed of $P$ at time $t$ seconds after leaving $O$ is $v \mathrm{~m} \mathrm{~s}^{-1}$, where $v$ is defined as follows.

$$ \begin{aligned} \text { For } 0 \leqslant t \leqslant 5, & v=t-0.1 t^{2}, \\ \text { for } 5 \leqslant t \leqslant 45, & v \text { is constant, } \\ \text { for } 45 \leqslant t \leqslant 50, & v=9 t-0.1 t^{2}-200. \end{aligned} $$

$\text{(i)}$ Find the distance travelled by $P$ in the first 5 seconds. $[3]$

$\text{(ii)}$ Find the total distance from $O$ to $A$, and deduce the average speed of $P$ for the whole journey from $O$ to $A$. $[6]$

Question 6 Code: 9709/43/O/N/13/5, Topic: -

A car travels in a straight line from $A$ to $B$, a distance of $12 \mathrm{~km}$, taking 552 seconds. The car starts from rest at $A$ and accelerates for $T_{1} \mathrm{~s}$ at $0.3 \mathrm{~m} \mathrm{~s}^{-2}$, reaching a speed of $V \mathrm{~m} \mathrm{~s}^{-1}$. The car then continues to move at $V \mathrm{~m} \mathrm{~s}^{-1}$ for $T_{2} \mathrm{~s}$. It then decelerates for $T_{3} \mathrm{~s}$ at $1 \mathrm{~ms}^{-2}$, coming to rest at $B$.

$\text{(i)}$ Sketch the velocity-time graph for the motion and express $T_{1}$ and $T_{3}$ in terms of $V$. $[3]$

$\text{(ii)}$ Express the total distance travelled in terms of $V$ and show that $13 V^{2}-3312 V+72000=0$. Hence find the value of $V$. $[5]$

Worked solutions: P1, P3 & P6 (S1)

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