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Name of student | Date | ||||

Adm. number | Year/grade | 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 | 5 | 5 | 7 | 9 | 8 | 6 | 10 | 9 | 8 | 8 | 11 | 14 | 100 |

Score |

Question 1 Code: 9709/41/M/J/16/1, Topic: -

A lift moves upwards from rest and accelerates at $0.9 \mathrm{~m} \mathrm{~s}^{-2}$ for $3 \mathrm{~s}$. The lift then travels for $6 \mathrm{~s}$ at constant speed and finally slows down, with a constant deceleration, stopping in a further $4 \mathrm{~s}$.

$\text{(i)}$ Sketch a velocity-time graph for the motion. $[3]$

$\text{(ii)}$ Find the total distance travelled by the lift. $[2]$

Question 2 Code: 9709/42/M/J/16/1, Topic: -

Coplanar forces of magnitudes $7 \mathrm{~N}, 6 \mathrm{~N}$ and $8 \mathrm{~N}$ act at a point in the directions shown in the diagram. Given that $\sin \alpha=\frac{3}{5}$, find the magnitude and direction of the resultant of the three forces. $[5]$

Question 3 Code: 9709/43/M/J/12/4, Topic: -

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 4 Code: 9709/43/M/J/11/5, Topic: -

A small block of mass $1.25 \mathrm{~kg}$ is on a horizontal surface. Three horizontal forces, with magnitudes and directions as shown in the diagram, are applied to the block. The angle $\theta$ is such that $\cos \theta=0.28$ and $\sin \theta=0.96$. A horizontal frictional force also acts on the block, and the block is in equilibrium.

$\text{(i)}$ Show that the magnitude of the frictional force is $7.5 \mathrm{~N}$ and state the direction of this force. $[4]$

$\text{(ii)}$ Given that the block is in limiting equilibrium, find the coefficient of friction between the block and the surface. $[2]$

The force of magnitude $6.1 \mathrm{~N}$ is now replaced by a force of magnitude $8.6 \mathrm{~N}$ acting in the same direction, and the block begins to move.

$\text{(iii)}$ Find the magnitude and direction of the acceleration of the block. $[3]$

Question 5 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 6 Code: 9709/43/M/J/17/5, Topic: -

A particle is projected vertically upwards from a point $O$ with a speed of $12 \mathrm{~ms}^{-1}$. Two seconds later a second particle is projected vertically upwards from $O$ with a speed of $20 \mathrm{~m} \mathrm{~s}^{-1}.$ At time $t \mathrm{~s}$ after the second particle is projected, the two particles collide.

$\text{(i)}$ Find $t$. $[5]$

$\text{(ii)}$ Hence find the height above $O$ at which the particles collide. $[1]$

Question 7 Code: 9709/41/M/J/19/5, Topic: -

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

$$ v=t^{2}-8 t+12 \quad \text { for } 0 \leqslant t \leqslant 8 $$$\text{(i)}$ Find the minimum velocity of $P$. $[3]$

$\text{(ii)}$ Find the total distance travelled by $P$ in the interval $0 \leqslant t \leqslant 8$. $[7]$

Question 8 Code: 9709/43/M/J/11/6, Topic: -

A lorry of mass $15000 \mathrm{~kg}$ climbs a hill of length $500 \mathrm{~m}$ at a constant speed. The hill is inclined at $2.5^{\circ}$ to the horizontal. The resistance to the lorry's motion is constant and equal to $800 \mathrm{~N}$.

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

On its return journey the lorry reaches the top of the hill with speed $20 \mathrm{~m} \mathrm{~s}^{-1}$ and continues down the hill with a constant driving force of $2000 \mathrm{~N}$. The resistance to the lorry's motion is again constant and equal to $800 \mathrm{~N}$.

$\text{(ii)}$ Find the speed of the lorry when it reaches the bottom of the hill. $[5]$

Question 9 Code: 9709/43/M/J/17/6, Topic: -

A car of mass $1200 \mathrm{~kg}$ is travelling along a horizontal road.

$\text{(i)}$ It is given that there is a constant resistance to motion.

$\text{(a)}$ The engine of the car is working at $16 \mathrm{~kW}$ while the car is travelling at a constant speed of $40 \mathrm{~m} \mathrm{~s}^{-1}$. Find the resistance to motion. $[2]$

$\text{(b)}$ The power is now increased to $22.5 \mathrm{~kW}$. Find the acceleration of the car at the instant it is travelling at a speed of $45 \mathrm{~m} \mathrm{~s}^{-1}$. $[3]$

$\text{(ii)}$ It is given instead that the resistance to motion of the car is $(590+2 v) \mathrm{N}$ when the speed of the car is $v \mathrm{~m} \mathrm{~s}^{-1}$. The car travels at a constant speed with the engine working at $16 \mathrm{~kW}$. Find this speed. $[3]$

Question 10 Code: 9709/42/M/J/18/6, Topic: -

A particle $P$ moves in a straight line passing through a point $O$. At time $t \mathrm{~s}$, the acceleration, $a \mathrm{~m} \mathrm{~s}^{-2}$, of $P$ is given by $a=6-0.24 t$. The particle comes to instantaneous rest at time $t=20$.

$\text{(i)}$ Find the value of $t$ at which the particle is again at instantaneous rest. $[5]$

$\text{(ii)}$ Find the distance the particle travels between the times of instantaneous rest. $[3]$

Question 11 Code: 9709/41/M/J/14/7, Topic: -

Two cyclists $P$ and $Q$ travel along a straight road $A B C$, starting simultaneously at $A$ and arriving simultaneously at $C$. Both cyclists pass through $B 400 \mathrm{~s}$ after leaving $A$. Cyclist $P$ starts with speed $3 \mathrm{~m} \mathrm{~s}^{-1}$ and increases this speed with constant acceleration $0.005 \mathrm{~m} \mathrm{~s}^{-2}$ until he reaches $B$.

$\text{(i)}$ Show that the distance $A B$ is $1600 \mathrm{~m}$ and find $P$ 's speed at $B$. $[3]$

Cyclist $Q$ travels from $A$ to $B$ with speed $v \mathrm{~m} \mathrm{~s}^{-1}$ at time $t$ seconds after leaving $A$, where

$$ v=0.04 t-0.0001 t^{2}+k \text {, } $$and $k$ is a constant.

$\text{(ii)}$ Find the value of $k$ and the maximum speed of $Q$ before he has reached $B$. $[6]$

Cyclist $P$ travels from $B$ to $C$, a distance of $1400 \mathrm{~m}$, at the speed he had reached at $B$. Cyclist $Q$ travels from $B$ to $C$ with constant acceleration $a \mathrm{~m} \mathrm{~s}^{-2}$.

$\text{(iii)}$ Find the time taken for the cyclists to travel from $B$ to $C$ and find the value of $a$. $[4]$

Question 12 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 $A$. $[9]$