Moderate -0.3 This is a multi-part mechanics question covering standard A-level topics (vectors, equilibrium, differential equations, friction, projectiles). Each part uses routine methods: (a) uses F=ma with vectors, (b) resolves forces in equilibrium, (c) is standard exponential growth, (d) compares applied force to limiting friction, (e) appears to be standard projectile motion. While it covers multiple topics, each requires straightforward application of formulas without novel insight, making it slightly easier than average.
An object of mass 4 kg is moving on a horizontal plane under the action of a constant force \(4 \mathbf { i } - 12 \mathbf { j } \mathrm {~N}\). At time \(t = 0 \mathrm {~s}\), its position vector is \(7 \mathbf { i } - 26 \mathbf { j }\) with respect to the origin \(O\) and its velocity vector is \(- \mathbf { i } + 4 \mathbf { j }\).
Determine the velocity vector of the object at time \(t = 5 \mathrm {~s}\).
Calculate the distance of the object from the origin when \(t = 2 \mathrm {~s}\).
The diagram below shows an object of weight 160 N at a point \(C\), supported by two cables \(A C\) and \(B C\) inclined at angles of \(23 ^ { \circ }\) and \(40 ^ { \circ }\) to the horizontal respectively.
\includegraphics[max width=\textwidth, alt={}, center]{b35e94ab-a426-4fca-9ecb-c659e0143ed7-5_444_919_973_612}
Find the tension in \(A C\) and the tension in \(B C\).
State two modelling assumptions you have made in your solution.
The rate of change of a population of a colony of bacteria is proportional to the size of the population \(P\), with constant of proportionality \(k\). At time \(t = 0\) (hours), the size of the population is 10 .
Find an expression, in terms of \(k\), for \(P\) at time \(t\).
Given that the population doubles after 1 hour, find the time required for the population to reach 1 million.
A particle of mass 12 kg lies on a rough horizontal surface. The coefficient of friction between the particle and the surface is 0.8 . The particle is at rest. It is then subjected to a horizontal tractive force of magnitude 75 N .
Determine the magnitude of the frictional force acting on the particle, giving a reason for your answer.
A body is projected at time \(t = 0 \mathrm {~s}\) from a point \(O\) with speed \(V \mathrm {~ms} ^ { - 1 }\) in a direction inclined at an angle of \(\theta\) to the horizontal.
Write down expressions for the horizontal and vertical components \(x \mathrm {~m}\) and \(y \mathrm {~m}\) of its displacement from \(O\) at time \(t \mathrm {~s}\).
Show that the range \(R \mathrm {~m}\) on a horizontal plane through the point of projection is given by
$$R = \frac { V ^ { 2 } } { g } \sin 2 \theta$$
Given that the maximum range is 392 m , find, correct to one decimal place,
i) the speed of projection,
ii) the time of flight,
iii) the maximum height attained.
\begin{enumerate}
\item An object of mass 4 kg is moving on a horizontal plane under the action of a constant force $4 \mathbf { i } - 12 \mathbf { j } \mathrm {~N}$. At time $t = 0 \mathrm {~s}$, its position vector is $7 \mathbf { i } - 26 \mathbf { j }$ with respect to the origin $O$ and its velocity vector is $- \mathbf { i } + 4 \mathbf { j }$.\\
(a) Determine the velocity vector of the object at time $t = 5 \mathrm {~s}$.\\
(b) Calculate the distance of the object from the origin when $t = 2 \mathrm {~s}$.
\item The diagram below shows an object of weight 160 N at a point $C$, supported by two cables $A C$ and $B C$ inclined at angles of $23 ^ { \circ }$ and $40 ^ { \circ }$ to the horizontal respectively.\\
\includegraphics[max width=\textwidth, alt={}, center]{b35e94ab-a426-4fca-9ecb-c659e0143ed7-5_444_919_973_612}\\
(a) Find the tension in $A C$ and the tension in $B C$.\\
(b) State two modelling assumptions you have made in your solution.
\item The rate of change of a population of a colony of bacteria is proportional to the size of the population $P$, with constant of proportionality $k$. At time $t = 0$ (hours), the size of the population is 10 .\\
(a) Find an expression, in terms of $k$, for $P$ at time $t$.\\
(b) Given that the population doubles after 1 hour, find the time required for the population to reach 1 million.
\item A particle of mass 12 kg lies on a rough horizontal surface. The coefficient of friction between the particle and the surface is 0.8 . The particle is at rest. It is then subjected to a horizontal tractive force of magnitude 75 N .\\
Determine the magnitude of the frictional force acting on the particle, giving a reason for your answer.
\item A body is projected at time $t = 0 \mathrm {~s}$ from a point $O$ with speed $V \mathrm {~ms} ^ { - 1 }$ in a direction inclined at an angle of $\theta$ to the horizontal.\\
(a) Write down expressions for the horizontal and vertical components $x \mathrm {~m}$ and $y \mathrm {~m}$ of its displacement from $O$ at time $t \mathrm {~s}$.\\
(b) Show that the range $R \mathrm {~m}$ on a horizontal plane through the point of projection is given by
\end{enumerate}
$$R = \frac { V ^ { 2 } } { g } \sin 2 \theta$$
(c) Given that the maximum range is 392 m , find, correct to one decimal place,\\
i) the speed of projection,\\
ii) the time of flight,\\
iii) the maximum height attained.
\hfill \mbox{\textit{WJEC Unit 4 Q6 [8]}}