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

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 | 4 | 3 | 6 | 5 | 5 | 6 | 6 | 8 | 6 | 10 | 8 | 8 | 75 |

Score |

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

A car of mass $700 \mathrm{~kg}$ is travelling along a straight horizontal road. The resistance to motion is constant and equal to $600 \mathrm{~N}$.

$\text{(i)}$ Find the driving force of the car's engine at an instant when the acceleration is $2 \mathrm{~m} \mathrm{~s}^{-2}$. $[2]$

$\text{(ii)}$ Given that the car's speed at this instant is $15 \mathrm{~m} \mathrm{~s}^{-1}$, find the rate at which the car's engine is working. $[2]$

Question 2 Code: 9709/42/O/N/13/1, Topic: -

A small block of weight $5.1 \mathrm{~N}$ rests on a smooth plane inclined at an angle $\alpha$ to the horizontal, where $\sin \alpha=\frac{8}{17}$. The block is held in equilibrium by means of a light inextensible string. The string makes an angle $\beta$ above the line of greatest slope on which the block rests, where $\sin \beta=\frac{7}{25}$ (see diagram). Find the tension in the string. $[3]$

Question 3 Code: 9709/43/M/J/19/2, Topic: -

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 4 Code: 9709/41/M/J/20/2, Topic: -

A car of mass $1800 \mathrm{~kg}$ is towing a trailer of mass $400 \mathrm{~kg}$ along a straight horizontal road. The car and trailer are connected by a light rigid tow-bar. The car is accelerating at $1.5 \mathrm{~m} \mathrm{~s}^{-2}.$ There are constant resistance forces of $250 \mathrm{~N}$ on the car and $100 \mathrm{~N}$ on the trailer.

$\text{(a)}$ Find the tension in the tow-bar. $[2]$

$\text{(b)}$ Find the power of the engine of the car at the instant when the speed is $20 \mathrm{~m} \mathrm{~s}^{-1}$. $[3]$

Question 5 Code: 9709/42/O/N/20/3, Topic: -

A block of mass $m \mathrm{~kg}$ is held in equilibrium below a horizontal ceiling by two strings, as shown in the diagram. One of the strings is inclined at $45^{\circ}$ to the horizontal and the tension in this string is $T \mathrm{~N}$. The other string is inclined at $60^{\circ}$ to the horizontal and the tension in this string is $20 \mathrm{~N}$.

Find $T$ and $m$. $[5]$

Question 6 Code: 9709/41/M/J/16/4, Topic: -

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 Code: 9709/42/O/N/16/4, Topic: -

A girl on a sledge starts, with a speed of $5 \mathrm{~m} \mathrm{~s}^{-1}$, at the top of a slope of length $100 \mathrm{~m}$ which is at an angle of $20^{\circ}$ to the horizontal. The sledge slides directly down the slope.

$\text{(i)}$ Given that there is no resistance to the sledge's motion, find the speed of the sledge at the bottom of the slope. $[3]$

$\text{(ii)}$ It is given instead that the sledge experiences a resistance to motion such that the total work done against the resistance is $8500 \mathrm{~J}$, and the speed of the sledge at the bottom of the slope is $21 \mathrm{~m} \mathrm{~s}^{-1}$. Find the total mass of the girl and the sledge. $[3]$

Question 8 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 9 Code: 9709/42/M/J/15/5, Topic: -

A particle $P$ starts from rest at a point $O$ on a horizontal straight line. $P$ moves along the line with constant acceleration and reaches a point $A$ on the line with a speed of $30 \mathrm{~m} \mathrm{~s}^{-1}$. At the instant that $P$ leaves $O$, a particle $Q$ is projected vertically upwards from the point $A$ with a speed of $20 \mathrm{~m} \mathrm{~s}^{-1}$. Subsequently $P$ and $Q$ collide at $A$. Find

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

$\text{(ii)}$ the distance $O A$. $[2]$

Question 10 Code: 9709/42/O/N/15/6, Topic: -

A small ring of mass $0.024 \mathrm{~kg}$ is threaded on a fixed rough horizontal rod. A light inextensible string is attached to the ring and the string is pulled with a force of magnitude $0.195 \mathrm{~N}$ at an angle of $\theta$ with the horizontal, where $\sin \theta=\frac{5}{13}$. When the angle $\theta$ is below the horizontal (see Fig. 1) the ring is in limiting equilibrium.

$\text{(i)}$ Find the coefficient of friction between the ring and the rod. $[6]$

When the angle $\theta$ is above the horizontal (see Fig. 2) the ring moves.

$\text{(ii)}$ Find the acceleration of the ring. $[4]$

Question 11 Code: 9709/42/M/J/16/6, Topic: -

A car of mass $1100 \mathrm{~kg}$ is moving on a road against a constant force of $1550 \mathrm{~N}$ resisting the motion.

$\text{(i)}$ The car moves along a straight horizontal road at a constant speed of $40 \mathrm{~m} \mathrm{~s}^{-1}$.

$\text{(a)}$ Calculate, in $\mathrm{kW}$, the power developed by the engine of the car. $[2]$

$\text{(b)}$ Given that this power is suddenly decreased by $22 \mathrm{~kW}$, find the instantaneous deceleration of the car. $[3]$

$\text{(ii)}$ The car now travels at constant speed up a straight road inclined at $8^{\circ}$ to the horizontal, with the engine working at $80 \mathrm{~kW}$. Assuming the resistance force remains the same, find this constant speed. $[3]$

Question 12 Code: 9709/42/O/N/18/6, Topic: -

A car of mass $1200 \mathrm{~kg}$ is driving along a straight horizontal road at a constant speed of $15 \mathrm{~m} \mathrm{~s}^{-1}$. There is a constant resistance to motion of $350 \mathrm{~N}$.

$\text{(i)}$ Find the power of the car's engine. $[1]$

The car comes to a hill inclined at $1^{\circ}$ to the horizontal, still travelling at $15 \mathrm{~m} \mathrm{~s}^{-1}$.

$\text{(ii)}$ The car starts to descend the hill with reduced power and with an acceleration of $0.12 \mathrm{~m} \mathrm{~s}^{-2}$. Given that there is no change in the resistance force, find the new power of the car's engine at the instant when it starts to descend the hill. $[3]$

$\text{(iii)}$ When the car is travelling at $20 \mathrm{~m} \mathrm{~s}^{-1}$ down the hill, the power is cut off and the car gradually slows down. Assuming that the resistance force remains $350 \mathrm{~N}$, find the distance travelled from the moment when the power is cut off until the speed of the car is reduced to $18 \mathrm{~m} \mathrm{~s}^{-1}$. $[4]$