MAST10006 Calculus 2, Semester 2, 2023
Assignment 8
School of Mathematics and Statistics, The University of Melbourne
Due by: 12pm (noon) Monday 9 October 2023
• Answer all the questions below. Of these questions, one will be chosen for marking.
• Submit your assignment in Canvas as a single PDF file before the deadline above.
See Canvas for instructions on how to submit the assignment.
• Marks may be awarded for:
◦ Correct use of appropriate mathematical techniques.
◦ Accuracy and validity of any calculations or algebraic manipulations.
◦ Clear justification or explanation of techniques and rules used.
◦ Clear communication of mathematical ideas through diagrams.
◦ Use of correct mathematical notation and terminology.
• You must use methods taught in MAST10006 Calculus 2 to solve the assignment questions.
• Give your answers as exact values.
1. Jumping on a trampoline can be modelled by a mass-spring system. A person is jumping
on a trampoline vertically and lightly so that his/her feet do not leave the trampoline.
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The jumper has a mass of m = 50 kg. The natural height of the trampoline is 1.5 m above
ground level and the trampoline has a spring constant k = 1000 N m−1. At equilibrium
the trampoline is compressed by a distance s metres.
Air resistance acts on the jumper with a damping constant β = 50 N s m−1. Assume that
the gravitational constant is g = 10 m s−2.
Let y(t) be the distance in metres of the jumper below the equilibrium position at time
t seconds after the start of the jump, and let h(t) be the height in metres of the jumper
above ground level at time t. At time t = 0, the jumper starts at height 0.6 m above the
ground and the velocity of the jumper is 0.
You may assume that the equation of motion for the system is
50y¨ + 50y˙ + 1000y = 0.
Please turn over
(a) Draw a diagram of the system when the jumper is below the equilibrium position
and moving up. Show all forces acting on the jumper and label them.
(b) Find s, and hence relate h(t) to y(t).
(c) Express the initial conditions in terms of y and y˙.
(d) Solve for y(t).
(e) Sketch h(t) versus t.
2. Suppose we now have a little ball bouncing vertically and lightly on the trampoline. The
ball’s mass is 0.5kg. Air resistance is unchanged. At time t = 0, the ball again starts at
height 0.6 m above the ground and the velocity of the ball is 0.
Draw a graph of h(t) in this case and briefly explain your reasoning.
Hint: you do not need to solve for y(t).
End of assignment