Modeling context with interpretation

A tourist decides to do a bungee jump from a bridge over a river. One end of an elastic rope is attached to the bridge and the other end of the elastic rope is attached to the tourist. The tourist jumps off the bridge. At time \(t\) seconds after the tourist reaches their lowest point, their vertical displacement is \(x\) metres above a fixed point 30 metres vertically above the river. When \(t = 0\)

• \(x = -20\)
• the velocity of the tourist is \(0\text{ms}^{-1}\)
• the acceleration of the tourist is \(13.6\text{ms}^{-2}\)

In the subsequent motion, the elastic rope is assumed to remain taut so that the vertical displacement of the tourist can be modelled by the differential equation $$5k\frac{d^2x}{dt^2} + 2k\frac{dx}{dt} + 17x = 0 \quad t \geq 0$$ where \(k\) is a positive constant.

1. Determine the value of \(k\) [2]
2. Determine the particular solution to the differential equation. [7]
3. Hence find, according to the model, the vertical height of the tourist above the river 15 seconds after they have reached their lowest point. [2]
4. Give a limitation of the model. [1]

The displacement of a door from its equilibrium (closed) position is measured by the angle, \(\theta\) radians, which the door makes with its closed position. The door can swing either side of the equilibrium position so that \(\theta\) can take positive and negative values. The door is released from rest from an open position at time \(t = 0\). A proposed differential equation to model the motion of the door for \(t \geqslant 0\) is $$\frac{\mathrm{d}^2\theta}{\mathrm{d}t^2} + \lambda \frac{\mathrm{d}\theta}{\mathrm{d}t} + 3\theta = 0$$ where \(\lambda\) is a constant and \(\lambda \geqslant 0\).

1. 1. According to the model, for what value of \(\lambda\) will the motion of the door be simple harmonic? [1]
   2. Explain briefly why modelling the motion of the door as simple harmonic is unlikely to be realistic. [1]
2. Find the range of values of \(\lambda\) for which the model predicts that the door will never pass through the equilibrium position. [2]
3. Sketch a possible graph of \(\theta\) against \(t\) when \(\lambda\) lies outside the range found in part (b) but the motion is not simple harmonic. [1]

A children's play centre has two rooms, a room full of bouncy castles and a room full of ball pits. At any given instant, each child in the centre is playing either on the bouncy castles or in the ball pits. Each child can see one room from the other room and can decide to change freely between the two rooms. It is assumed that such changes happen instantaneously. The number of children playing on the bouncy castles at time \(t\) hours, is denoted by \(C\) and the corresponding number of children playing in the ball pits is \(P\). Because the number of children is large for most of the time, \(C\) and \(P\) are modelled as being continuous. When there is a different number of children in each room, some children will move from the room with more children to the room with fewer children. A researcher therefore decides to model \(C\) and \(P\) with the following coupled differential equations. $$\frac{dP}{dt} = \alpha(P-C) + \gamma t$$ $$\frac{dC}{dt} = \alpha(C-P)$$

1. Explain why \(\alpha\) must be negative. [1]

After examining data, the researcher chooses \(\alpha = -2\) and \(\gamma = 32\).

1. Show that \(P\) satisfies the second order differential equation \(\frac{d^2P}{dt^2} + 4\frac{dP}{dt} = 64t + 32\). [2]
2. 1. Find the complementary function for the differential equation from part (b). [1]
   2. Explain why a particular integral of the form \(P = at + b\) will not work in this situation. [1]
   3. Using a particular integral of the form \(P = at^2 + bt\), find the general solution of the differential equation from part (b). [3]

At a certain time there are 55 children playing in the ball pits and 24 children per hour are arriving at the ball pits.

1. Use the model, starting from this time, to estimate the number of children in the ball pits 30 minutes later. [4]
2. Explain why the model becomes unreliable as \(t\) gets very large. [1]

During an industrial process substance \(X\) is converted into substance \(Z\). Some of the substance \(X\) goes through an intermediate phase, and is converted to substance \(Y\), before being converted to substance \(Z\). The situation is modelled by $$\frac{dy}{dt} = 0.3x + 0.2y \text{ and } \frac{dz}{dt} = 0.2y + 0.1x$$ where \(x\), \(y\) and \(z\) are the amounts in kg of \(X\), \(Y\) and \(Z\) at time \(t\) hours after the process starts. Initially there is 10 kg of substance \(X\) and nothing of substances \(Y\) and \(Z\). The amount of substance \(X\) decreases exponentially. The initial rate of decrease is 4 kg per hour.

1. Show that \(x = Ae^{-0.4t}\), stating the value of \(A\). [3]
2. 1. Show that \(\frac{dx}{dt} + \frac{dy}{dt} + \frac{dz}{dt} = 0\). [2]
   2. Comment on this result in the context of the industrial process. [2]
3. Express \(y\) in terms of \(t\). [5]
4. Determine the maximum amount of substance \(Y\) present during the process. [3]
5. How long does it take to produce 9 kg of substance \(Z\)? [2]

Two species of insects, \(X\) and \(Y\), co-exist on an island. The populations of the species at time \(t\) years are \(x\) and \(y\) respectively, where \(x\) and \(y\) are measured in millions. The situation can be modelled by the differential equations $$\frac{\mathrm{d}x}{\mathrm{d}t} = 3x + 10y,$$ $$\frac{\mathrm{d}y}{\mathrm{d}t} = x + 6y.$$

1. 1. Show that \(\frac{\mathrm{d}^2x}{\mathrm{d}t^2} - 9\frac{\mathrm{d}x}{\mathrm{d}t} + 8x = 0\).
   2. Find the general solution for \(x\) in terms of \(t\). [7]
2. Find the corresponding general solution for \(y\). [4]
3. When \(t = 0\), \(\frac{\mathrm{d}x}{\mathrm{d}t} = 5\) and the population of species \(Y\) is 4 times the population of species \(X\). Find the particular solution for \(x\) in terms of \(t\). [6]

A capacitor is an electrical component which stores charge. The value of the charge stored by the capacitor, in suitable units, is denoted by \(Q\). The capacitor is placed in an electrical circuit. At any time \(t\) seconds, where \(t \geq 0\), \(Q\) can be modelled by the differential equation $$\frac{d^2Q}{dt^2} - 2\frac{dQ}{dt} - 15Q = 0.$$ Initially the charge is 100 units and it is given that \(Q\) tends to a finite limit as \(t\) tends to infinity.

1. Determine the charge on the capacitor when \(t = 0.5\). [6]
2. Determine the finite limit of \(Q\) as \(t\) tends to infinity. [1]

The quantity of grass on an island at time \(t\) years is \(x\), in appropriate units. At time \(t = 0\) some rabbits are introduced to the island. The population of rabbits on the island at time \(t\) years is \(y\), in units of \(100\)s of rabbits. An ecologist who is studying the island suggests that the following pair of simultaneous first order differential equations can be used to model the population of rabbits and quantity of grass for \(t \geq 0\). $$\frac{dx}{dt} = 3x - 2y,$$ $$\frac{dy}{dt} = y + 5x$$

1. 1. Show that \(\frac{d^2x}{dt^2} = a\frac{dx}{dt} + bx\) where \(a\) and \(b\) are constants which should be found. [2]
   2. Find the general solution for \(x\) in real form. [3]
2. Find the corresponding general solution for \(y\). [3]

At time \(t = 0\) the quantity of grass on the island was \(4\) units. The number of rabbits introduced at this time was \(500\).

1. Find the particular solutions for \(x\) and \(y\). [5]
2. The ecologist finds that the model predicts that there will be no grass at time \(T\), when there are still rabbits on the island. Find the value of \(T\). [1]
3. State one way in which the model is not appropriate for modelling the quantity of grass and the population of rabbits for \(0 \leq t \leq T\). [1]

A swing door is a door to a room which is closed when in equilibrium but which can be pushed open from either side and which can swing both ways, into or out of the room, and through the equilibrium position. The door is sprung so that when displaced from the equilibrium position it will swing back towards it. The extent to which the door is open at any time, \(t\) seconds, is measured by the angle at the hinge, \(\theta\), which the plane of the door makes with the plane of the equilibrium position. See the diagram below. In an initial model of the motion of a certain swing door it is suggested that \(\theta\) satisfies the following differential equation. $$4\frac{\mathrm{d}^2\theta}{\mathrm{d}t^2} + 25\theta = 0 \quad (*)$$

1. 1. Write down the general solution to (*). [2]
   2. With reference to the behaviour of your solution in part (a)(i) explain briefly why the model using (*) is unlikely to be realistic. [1]

In an improved model of the motion of the door an extra term is introduced to the differential equation so that it becomes $$4\frac{\mathrm{d}^2\theta}{\mathrm{d}t^2} + \lambda\frac{\mathrm{d}\theta}{\mathrm{d}t} + 25\theta = 0 \quad (\dagger)$$ where \(\lambda\) is a positive constant.

1. In the case where \(\lambda = 16\) the door is held open at an angle of \(0.9\) radians and then released from rest at time \(t = 0\).
   1. Find, in a real form, the general solution of (\(\dagger\)). [3]
   2. Find the particular solution of (\(\dagger\)). [4]
   3. With reference to the behaviour of your solution found in part (b)(ii) explain briefly how the extra term in (\(\dagger\)) improves the model. [2]
2. Find the value of \(\lambda\) for which the door is critically damped. [2]