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PAPER CODE NO.
MATH 122
MAY EXAMINATIONS 2008
Bachelor of Science : Year 1
Bachelor of Science : Year 2
Master of Mathematics : Year 1
Master of Mathematics : Year 2
Master of Physics : Year 1
DYNAMIC MODELLING
TIME ALLOWED: Two Hours and a Half
INSTRUCTIONS TO CANDIDATES
You may attempt all questions. All answers to Section A and the best THREE answers to questions from Section B will be taken into account. Section A carries 55% of the available marks.
Take gravitational acceleration ms-2^ unless stated otherwise. Give numerical answers to 3 significant digits.
SECTION A
- A pizza baked at 200°C is removed from the oven and placed on the kitchen sideboard to cool. The temperature of the pizza at time is decreasing at a rate proportional to the difference between the ambient room temperature (20°C) and , with constant of proportionality s-1. Write down a differential equation for , and show that
How many minutes does it take for the pizza to cool down to 40°C? [6 marks]
- A bridge road has only one lane. Traffic lights are installed on both sides of the bridge so that traffic uses the lane, alternating in direction. Vehicles pass through a green light at a rate of 12 vehicles per minute. Vehicles arriving at the first set of lights, which are green for minutes, do so at a rate of 4 per minute. Vehicles moving in opposite direction arrive at the second set of lights, which are green for minutes, at a rate of 6 per minute. Both sets of lights are red after each green light for 1 minute in order for the lane to clear. Find conditions for and to avoid congestion in each of the directions. Shade the region in the plane for the times satisfied by these inequalities. Find the minimum values for and. [6 marks]
- Let and , where denotes time, represent the levels of two fish populations in a pond. Suppose that these satisfy the differential equations
Initially (at ) and. Obtain and solve the differential equation for in terms of. From your results draw a phase diagram for this situation, indicating which way around the curve the point moves. [7 marks]
- A ball of mass , initially at rest, is dropped vertically from a tower 40 metres high. As it falls the ball is subject to a resistive force equivalent to N, where is its velocity at time. Write down Newton’s equation of motion and show that
Integrate this expression exactly to find the distance travelled by the ball at time t. Using approximation (for ), estimate the time when the ball hits the ground. [9 marks]
- A spring of negligible mass, when suspended vertically from one end, is stretched a distance of 0.1 metres when a mass is added. Calculate its spring constant (take ms-2). The unstretched spring is placed horizontally on a rough table, with the mass attached to one end, the other being firmly fixed to a nearby wall. The mass is then pulled away from the wall a further 0.2 metres and released from rest. Assuming that friction imposes a damping force equal to times the instantaneous speed, show that the differential equation describing the motion of the mass is
where represents the extension of the spring. Solve this equation.
[9 marks]
SECTION B
- Consider a two-state stochastic system, with states A and B. In the usual notations,
Briefly explain what this equation represents. Derive an equation for the rate of change of. [5 mark] A binary computer code is represented by a sequence of “1”s and “0”s. For a certain code the following correlations have been found: (1) the probability of “1” followed by “1” is 0.7; (2) the probability of “0” followed by “0” is 0.6. Write down the equation for , hence show that
Solve this equation to find , given that , and find the long- term value of. [10 marks]
- A model for the number of predators at time uses the equation
where is the number of preys at time , which satisfies the equation:
Find the equation for and integrate it, given that initially, at time , and. [6 marks] Sketch the graph of against indicating the direction and change over time. Describe what happens to the two populations over time. Calculate the maximum prey population before it starts to become extinct. [9 marks]
- Suppose that the size of an animal population is changing at a rate of where is the size of the population at time t. Write down the differential equation for. What are the equilibrium values of and which of them is stable. [8 marks] Solve the differential equation to find given that there were 20 animals at time . [Hint: use partial fractions] [7 marks]
