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.