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DEXTERSMITHNUS

Cambridge International AS and A Level

Name of student DEXTERSMITHNUS Date
Adm. number Year/grade 1984 Stream DexterSmithnus
Subject Mechanics 1 (M1) Variant(s) P41, P42, P43
Start time Duration Stop time

Qtn No. 1 2 3 4 5 6 Total
Marks 12 14 13 12 11 10 72
Score

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Question 1 Code: 9709/41/M/J/18/7, Topic: -

 

The diagram shows a triangular block with sloping faces inclined to the horizontal at $45^{\circ}$ and $30^{\circ}$. Particle $A$ of mass $0.8 \mathrm{~kg}$ lies on the face inclined at $45^{\circ}$ and particle $B$ of mass $1.2 \mathrm{~kg}$ lies on the face inclined at $30^{\circ}$. The particles are connected by a light inextensible string which passes over a small smooth pulley $P$ fixed at the top of the faces. The parts $A P$ and $B P$ of the string are parallel to lines of greatest slope of the respective faces. The particles are released from rest with both parts of the string taut. In the subsequent motion neither particle reaches the pulley and neither particle reaches the bottom of a face.

$\text{(i)}$ Given that both faces are smooth, find the speed of $A$ after each particle has travelled a distance of $0.4 \mathrm{~m}$. $[6]$

$\text{(ii)}$ It is given instead that both faces are rough. The coefficient of friction between each particle and a face of the block is $\mu$. Find the value of $\mu$ for which the system is in limiting equilibrium. $[6]$

Question 2 Code: 9709/42/M/J/18/7, Topic: -

 

As shown in the diagram, a particle $A$ of mass $1.6 \mathrm{~kg}$ lies on a horizontal plane and a particle $B$ of mass $2.4 \mathrm{~kg}$ lies on a plane inclined at an angle of $30^{\circ}$ to the horizontal. The particles are connected by a light inextensible string which passes over a small smooth pulley $P$ fixed at the top of the inclined plane. The distance $A P$ is $2.5 \mathrm{~m}$ and the distance of $B$ from the bottom of the inclined plane is $1 \mathrm{~m}$. There is a barrier at the bottom of the inclined plane preventing any further motion of $B$. The part $B P$ of the string is parallel to a line of greatest slope of the inclined plane. The particles are released from rest with both parts of the string taut.

$\text{(i)}$ Given that both planes are smooth, find the acceleration of $A$ and the tension in the string. $[5]$

$\text{(ii)}$ It is given instead that the horizontal plane is rough and that the coefficient of friction between $A$ and the horizontal plane is $0.2$. The inclined plane is smooth. Find the total distance travelled by $A$. $[9]$

Question 3 Code: 9709/43/M/J/18/7, Topic: -

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

$$ \begin{array}{ll} v=12 t-4 t^{2} & \text { for } 0 \leqslant t \leqslant 2 \\ v=16-4 t & \text { for } 2 \leqslant t \leqslant 4 \end{array} $$

$\text{(i)}$ Find the maximum velocity of $P$ during the first $2 \mathrm{~s}$. $[3]$

$\text{(ii)}$ Determine, with justification, whether there is any instantaneous change in the acceleration of $P$ when $t=2$. $[2]$

$\text{(iii)}$ Sketch the velocity-time graph for $0 \leqslant t \leqslant 4$. $[3]$

$\text{(iv)}$ Find the distance travelled by $P$ in the interval $0 \leqslant t \leqslant 4 .$

Question 4 Code: 9709/41/O/N/18/7, Topic: -

A particle moves in a straight line starting from rest from a point $O$. The acceleration of the particle at time $t \mathrm{~s}$ after leaving $O$ is $a \mathrm{~m} \mathrm{~s}^{-2}$, where

$$ a=5.4-1.62 t $$

$\text{(i)}$ Find the positive value of $t$ at which the velocity of the particle is zero, giving your answer as an exact fraction. $[4]$

$\text{(ii)}$ Find the velocity of the particle at $t=10$ and sketch the velocity-time graph for the first ten seconds of the motion. $[3]$

$\text{(iii)}$ Find the total distance travelled during the first ten seconds of the motion. $[5]$

Question 5 Code: 9709/42/O/N/18/7, Topic: -

A particle of mass $0.3 \mathrm{~kg}$ is released from rest above a tank containing water. The particle falls vertically, taking $0.8 \mathrm{~s}$ to reach the water surface. There is no instantaneous change of speed when the particle enters the water. The depth of water in the tank is $1.25 \mathrm{~m}$. The water exerts a force on the particle resisting its motion. The work done against this resistance force from the instant that the particle enters the water until it reaches the bottom of the tank is $1.2 \mathrm{~J}$.

$\text{(i)}$ Use an energy method to find the speed of the particle when it reaches the bottom of the tank. $[4]$

When the particle reaches the bottom of the tank, it bounces back vertically upwards with initial speed $7 \mathrm{~m} \mathrm{~s}^{-1}$. As the particle rises through the water, it experiences a constant resistance force of $1.8 \mathrm{~N}$. The particle comes to instantaneous rest $t$ seconds after it bounces on the bottom of the tank.

$\text{(ii)}$ Find the value of $t$. $[7]$

Question 6 Code: 9709/43/O/N/18/7, Topic: -

A particle moves in a straight line. The particle is initially at rest at a point $O$ on the line. At time $t \mathrm{~s}$ after leaving $O$, the acceleration $a \mathrm{~m} \mathrm{~s}^{-2}$ of the particle is given by $a=25-t^{2}$ for $0 \leqslant t \leqslant 9$.

$\text{(i)}$ Find the maximum velocity of the particle in this time period. $[4]$

$\text{(ii)}$ Find the total distance travelled until the maximum velocity is reached. $[2]$

The acceleration of the particle for $t>9$ is given by $a=-3 t^{-\frac{1}{2}}$.

$\text{(iii)}$ Find the velocity of the particle when $t=25$. $[4]$

Worked solutions: P1, P3 & P6 (S1)

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