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OTIA - AS MATHEMATICS

Cambridge International AS and A Level

Name of student Date
Adm. number Year/grade 11 Stream
Subject Mechanics 1 (M1) Variant(s) P41, P42, P43
Start time Duration Stop time

Qtn No. 1 2 3 4 5 6 7 8 9 10 11 12 Total
Marks 4 6 7 6 7 6 7 7 6 11 10 9 86
Score

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Question 1

A particle of mass $8 \mathrm{~kg}$ is pulled at a constant speed a distance of $20 \mathrm{~m}$ up a rough plane inclined at an angle of $30^{\circ}$ to the horizontal by a force acting along a line of greatest slope.

$\text{(i)}$ Find the change in gravitational potential energy of the particle. $[2]$

$\text{(ii)}$ The total work done against gravity and friction is $1146 \mathrm{~J}$. Find the frictional force acting on the particle. $[2]$

Question 2

 

Coplanar forces of magnitudes $12 \mathrm{~N}, 24 \mathrm{~N}$ and $30 \mathrm{~N}$ act at a point in the directions shown in the diagram.

$\text{(i)}$ Find the components of the resultant of the three forces in the $x$-direction and in the $y$-direction. $[4]$

Component in $x$-direction.

Component in $y$-direction.

$\text{(ii)}$ Hence find the direction of the resultant. $[2]$

Question 3

 

Four coplanar forces act at a point. The magnitudes of the forces are $5 \mathrm{~N}, 4 \mathrm{~N}, 3 \mathrm{~N}$ and $7 \mathrm{~N}$, and the directions in which the forces act are shown in the diagram. Find the magnitude and direction of the resultant of the four forces. $[6]$

Question 4

A particle of mass $8 \mathrm{~kg}$ is projected with a speed of $5 \mathrm{~m} \mathrm{~s}^{-1}$ up a line of greatest slope of a rough plane inclined at an angle $\alpha$ to the horizontal, where $\sin \alpha=\frac{5}{13}$. The motion of the particle is resisted by a constant frictional force of magnitude $15 \mathrm{~N}$. The particle comes to instantaneous rest after travelling a distance $x \mathrm{~m}$ up the plane.

$\text{(i)}$ Express the change in gravitational potential energy of the particle in terms of $x$. $[2]$

$\text{(ii)}$ Use an energy method to find $x$. $[4]$

Question 5

A car of mass $1230 \mathrm{~kg}$ increases its speed from $4 \mathrm{~m} \mathrm{~s}^{-1}$ to $21 \mathrm{~m} \mathrm{~s}^{-1}$ in $24.5 \mathrm{~s}$. The table below shows corresponding values of time $t \mathrm{~s}$ and speed $v \mathrm{~m} \mathrm{~s}^{-1}$.

$$ \begin{array}{|c|c|c|c|c|} \hline t & 0 & 0.5 & 16.3 & 24.5 \\ \hline v & 4 & 6 & 19 & 21 \\ \hline \end{array} $$

$\text{(i)}$ Using the values in the table, find the average acceleration of the car for $0< t <0.5$ and for $16.3< t <24.5.$ $[2]$

While the car is increasing its speed the power output of its engine is constant and equal to $P \mathrm{~W}$, and the resistance to the car's motion is constant and equal to $R \mathrm{~N}$.

$\text{(ii)}$ Assuming that the values obtained in part $\text{(i)}$ are approximately equal to the accelerations at $v=5$ and at $v=20$, find approximations for $P$ and $R$. $[5]$

Question 6

 

Coplanar forces of magnitudes $50 \mathrm{~N}, 48 \mathrm{~N}, 14 \mathrm{~N}$ and $P \mathrm{~N}$ act at a point in the directions shown in the diagram. The system is in equilibrium. Given that tan $\alpha=\frac{7}{24}$, find the values of $P$ and $\theta$. $[6]$

Question 7

Question 8

A particle $P$ moves in a straight line $A B C D$ with constant deceleration. The velocities of $P$ at $A, B$ and $C$ are $20 \mathrm{~m} \mathrm{~s}^{-1}, 12 \mathrm{~m} \mathrm{~s}^{-1}$ and $6 \mathrm{~m} \mathrm{~s}^{-1}$ respectively.

$\text{(i)}$ Find the ratio of distances $A B: B C$. $[4]$

$\text{(ii)}$ The particle comes to rest at $D$. Given that the distance $A D$ is $80 \mathrm{~m}$, find the distance $B C$. $[3]$

Question 9

A particle of mass $3 \mathrm{~kg}$ is on a rough plane inclined at an angle of $20^{\circ}$ to the horizontal. A force of magnitude $P \mathrm{~N}$ acting parallel to a line of greatest slope of the plane is used to keep the particle in equilibrium. The coefficient of friction between the particle and the plane is $0.35$. Show that the least possible value of $P$ is $0.394$, correct to 3 significant figures, and find the greatest possible value of $P$. $[6]$

Question 10

Question 11

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 12

 

Two rectangular boxes $A$ and $B$ are of identical size. The boxes are at rest on a rough horizontal floor with $A$ on top of $B$. Box $A$ has mass $200 \mathrm{~kg}$ and box $B$ has mass $250 \mathrm{~kg}$. A horizontal force of magnitude $P \mathrm{~N}$ is applied to $B$ (see diagram). The boxes remain at rest if $P \leqslant 3150$ and start to move if $P>3150$.

$\text{(i)}$ Find the coefficient of friction between $B$ and the floor. $[3]$

The coefficient of friction between the two boxes is $0.2$. Given that $P>3150$ and that no sliding takes place between the boxes,

$\text{(ii)}$ show that the acceleration of the boxes is not greater than $2 \mathrm{~m} \mathrm{~s}^{-2}$, $[3]$

$\text{(iii)}$ find the maximum possible value of $P$. $[3]$

Worked solutions: P1, P3 & P6 (S1)

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