4.10b Model with differential equations: kinematics and other contexts

A rocket, with initial mass 1500 kg, including 600 kg of fuel, is launched vertically upwards from rest. The rocket burns fuel at a rate of 15 kg s\(^{-1}\) and the burnt fuel is ejected vertically downwards with a speed of 1000 m s\(^{-1}\) relative to the rocket. At time \(t\) seconds after launch \((t \leqslant 40)\) the rocket has mass \(m\) kg and velocity \(v\) m s\(^{-1}\).

1. Show that $$\frac{dv}{dt} + \frac{1000}{m}\frac{dm}{dt} = -9.8$$ [5]
2. Find \(v\) at time \(t\), \(0 \leqslant t \leqslant 40\) [5]

A spacecraft is travelling in a straight line in deep space where all external forces can be assumed to be negligible. The spacecraft decelerates by ejecting fuel at a constant speed \(k\) relative to the spacecraft, in the direction of motion of the spacecraft. At time \(t\), the spacecraft has speed \(v\) and mass \(m\).

1. Show, from first principles, that while the spacecraft is ejecting fuel, $$\frac{dv}{dm} - \frac{k}{m} = 0$$ [5]

At time \(t = 0\), the spacecraft has speed \(U\) and mass \(M\).

1. Find the mass of the spacecraft when it comes to rest. [6]

Given that \(m = Me^{-\alpha t^2}\), where \(\alpha\) is a positive constant, and that the spacecraft comes to rest at time \(t = T\),

1. find, in terms of \(U\) and \(T\) only, the distance travelled by the spacecraft in decelerating from speed \(U\) to rest. [6]

An isolated island is populated by rabbits and foxes. At time \(t\) the number of rabbits is \(x\) and the number of foxes is \(y\). It is assumed that: • The number of foxes increases at a rate proportional to the number of rabbits. When there are 200 rabbits the number of foxes is increasing at a rate of 20 foxes per unit period of time. • If there were no foxes present, the number of rabbits would increase by 120% in a unit period of time. • When both foxes and rabbits are present the foxes kill rabbits at a rate that is equal to 110% of the current number of foxes. • At time \(t = 0\), the number of foxes is 20 and the number of rabbits is 80.

1. 1. Construct a mathematical model for the number of rabbits. [9 marks]
   2. Use this model to show that the number of rabbits has doubled after approximately 0.7 units of time. [1 mark]
2. Suggest one way in which the model that you have used for the number of rabbits could be refined. [1 mark]

\includegraphics{figure_15} Two tanks, A and B, each have a capacity of 800 litres. At time \(t = 0\) both tanks are full of pure water. When \(t > 0\), water flows in the following ways: • Water with a salt concentration of \(\mu\) grams per litre flows into tank A at a constant rate • Water flows from tank A to tank B at a rate of 16 litres per minute • Water flows from tank B to tank A at a rate of \(r\) litres per minute • Water flows out of tank B through a waste pipe • The amount of water in each tank remains at 800 litres. At time \(t\) minutes (\(t \geq 0\)) there are \(x\) grams of salt in tank A and \(y\) grams of salt in tank B. This system is represented by the coupled differential equations \begin{align} \frac{dx}{dt} &= 36 - 0.02x + 0.005y \tag{1}
\frac{dy}{dt} &= 0.02x - 0.02y \tag{2} \end{align}

1. Find the value of \(r\). [2 marks]
2. Show that \(\mu = 3\) [3 marks]
3. Solve the coupled differential equations to find both \(x\) and \(y\) in terms of \(t\). [9 marks]

A population of meerkats is being studied. The population is modelled by the differential equation $$\frac{\mathrm{d}P}{\mathrm{d}t} = \frac{1}{22}P(11 - 2P), \quad t \geq 0, \quad 0 < P < 5.5$$ where \(P\), in thousands, is the population of meerkats and \(t\) is the time measured in years since the study began. Given that there were 1000 meerkats in the population when the study began, determine the time taken, in years, for this population of meerkats to double. [7]

A stone of mass \(m\) is projected vertically upwards with initial velocity \(u\). At time \(t\), the height risen above the point of projection is \(x\) and the resistance to motion is \(kv\) when the velocity of the stone is \(v\).

1. Write down a first-order differential equation relating \(v\) and \(t\) and hence find \(t\) in terms of \(v\). [5]
2. Write down a first-order differential equation relating \(v\) and \(x\) and hence find \(x\) in terms of \(v\). [6]