Small oscillations with non-standard force laws

4 A particle \(P\) of mass \(m\) moves along part of a horizontal straight line \(A B\). The mid-point of \(A B\) is \(O\), and \(A B = 4 a\). At time \(t , A P = 2 a + x\). The particle \(P\) is acted on by two horizontal forces. One force has magnitude \(m g \left( \frac { 2 a + x } { 2 a } \right) ^ { - \frac { 1 } { 2 } }\) and acts towards \(A\); the other force has magnitude \(m g \left( \frac { 2 a - x } { 2 a } \right)\) and acts towards \(B\). Show that, provided \(\frac { x } { a }\) remains small, \(P\) moves in approximate simple harmonic motion with centre \(O\), and state the period of this motion. At time \(t = 0 , P\) is released from rest at the point where \(x = \frac { 1 } { 20 } a\). Show that the speed of \(P\) when \(x = \frac { 1 } { 40 } a\) is \(\frac { 1 } { 80 } \sqrt { } ( 3 a g )\), and find the value of \(t\) when \(P\) reaches this point for the first time.

5 A particle of mass \(m\) moves in a straight line \(A B\) of length \(2 a\). When the particle is at a general point \(P\) there are two forces acting, one in the direction \(\overrightarrow { P A }\) with magnitude \(m g \left( \frac { P A } { a } \right) ^ { - \frac { 1 } { 4 } }\) and the other in the direction \(\overrightarrow { P B }\) with magnitude \(m g \left( \frac { P B } { a } \right) ^ { \frac { 1 } { 2 } }\). At time \(t = 0\) the particle is released from rest at the point \(C\), where \(A C = 1.04 a\). At time \(t\) the distance \(A P\) is \(a + x\). Show that the particle moves in approximate simple harmonic motion. Using the approximate simple harmonic motion, find the speed of \(P\) when it first reaches the mid-point of \(A B\) and the time taken for \(P\) to first reach half of this speed.

6. \begin{figure}[h] \end{figure} A river is 30 m wide and flows between two straight parallel banks. At each point of the river, the direction of flow is parallel to the banks. At time \(t = 0\), a boat leaves a point \(O\) on one bank and moves in a straight line across the river to a point \(P\) on the opposite bank. Its path \(O P\) is perpendicular to both banks and \(O P = 30 \mathrm {~m}\), as shown in Figure 2. The speed of flow of the river, \(r \mathrm {~m} \mathrm {~s} ^ { - 1 }\), at a point on \(O P\) which is at a distance \(x \mathrm {~m}\) from \(O\), is modelled as $$r = \frac { 1 } { 10 } x , \quad 0 \leq x \leq 30$$ The speed of the boat relative to the water is constant at \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). At time \(t\) seconds the boat is at a distance \(x \mathrm {~m}\) from \(O\) and is moving with speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in the direction \(O P\).

1. Show that $$100 v ^ { 2 } = 2500 - x ^ { 2 }$$
2. Hence show that $$\frac { \mathbf { d } ^ { 2 } x } { \mathbf { d } t ^ { 2 } } + \frac { x } { 100 } = 0$$
3. Find the total time taken for the boat to cross the river from \(O\) to \(P\).
   (9)

4
\includegraphics[max width=\textwidth, alt={}, center]{6e3d5f5e-7ffa-4111-903d-468fb4d20192-2_364_1313_1224_376} An unidentified aircraft \(U\) is flying horizontally with constant velocity \(250 \mathrm {~ms} ^ { - 1 }\) in the direction with bearing \(040 ^ { \circ }\). Two spotter planes \(P\) and \(Q\) are flying horizontally at the same height as \(U\), and at one instant \(P\) is 15000 m due west of \(U\), and \(Q\) is 15000 m due east of \(U\) (see diagram).

1. Plane \(P\) is flying with constant velocity \(210 \mathrm {~ms} ^ { - 1 }\) in the direction with bearing \(070 ^ { \circ }\).

4 In this question you are required to consider the family of differential equations $$\frac { d P } { d t } = r P \left( 1 - \frac { P } { K } \right) , \quad t \geqslant 0 , \quad P ( t ) \geqslant 0 \left( ^ { * } \right)$$ where \(r\) and \(K\) are positive constants. This differential equation can be used as a model for the size of a population \(P\) as a function of time \(t\).

1. 1. Determine the values of \(P\) for which
      • \(\frac { \mathrm { dP } } { \mathrm { dt } } = 0\)
2. \(\frac { \mathrm { dP } } { \mathrm { dt } } > 0\)
3. \(\frac { \mathrm { dP } } { \mathrm { dt } } < 0\)
   (ii) Solve the equation (*) subject to the initial condition that \(P = P _ { 0 }\) when \(t = 0\).
   (iii) Find a property common to your solution in (ii) in the cases \(\mathrm { P } _ { 0 } > \mathrm { K }\) and \(\mathrm { P } _ { 0 } < \mathrm { K }\).
   (iv) State a feature of your solution in (iii) for the case \(\mathrm { P } _ { 0 } > \mathrm { K }\) which is different to the case \(P _ { 0 } < K\).
   (v) Interpret the value \(K\) when \(P ( t )\) is the size of a population at time \(t\).
4. In this question you will explore the limitations of using the Euler method to approximate solutions to the differential equation
$$\frac { d P } { d t } = 2 P ^ { 1.25 } \left( 1 - \frac { P } { 1000 } \right) ^ { 1.5 } , t \geqslant 0 , P ( t ) \geqslant 0 ( * * )$$ The diagram shows the tangent field to (**), and a solution in which \(P = 1\) when \(t = 0\), produced using a much more accurate numerical method.
\includegraphics[max width=\textwidth, alt={}, center]{4715d0f0-a860-4189-802f-1d2d019e1115-4_899_1552_1763_319}
(i) The Euler method for the solution of the differential equation \(f ( t , P ) = \frac { d P } { d t }\) is as follows $$P _ { n + 1 } = P _ { n } + h f \left( t _ { n } , P _ { n } \right)$$ It is given that \(t _ { 0 } = 0\) and \(P _ { 0 } = 1\).
• Construct a spreadsheet to solve (**) using the Euler method so that the value of \(h\) can be varied.
• State the formulae you have used in your spreadsheet.
  (ii) Use your spreadsheet with \(h = 0.1\) to approximate
• \(P ( 1 )\)
• \(P ( 2 )\)
• \(P ( 3 )\)
  (iii) Use your spreadsheet with \(h = 0.05\) to approximate
• \(P ( 1 )\)
• \(P ( 2 )\)
• \(P ( 3 )\)
  (iv) State, with reasons, whether the estimates to \(P ( t )\) given in your spreadsheet are likely to be overestimates or underestimates to the exact values.
  (v) With reference to the diagram, explain any noticeable feature identified when comparing the approximations given to \(P ( 2 )\) in (ii) and (iii).