Hello, dear friend, you can consult us at any time if you have any questions, add  WeChat:  zz-x2580

Task 1 Read Section 14.6 Numerical Solutions of an old version of Bedford & Fowler’s Engineering
Mechanics text. You may have this book from your Applied Mechanics course or find it
in the library. In case the numbering is different in your copy of the text, Chapter 14 is
labelled “Force, Mass and Acceleration”.
Task 2 Show that the state of motion of the object at t = t0 + ∆t (i.e. a short time after some
arbitrary time t0) can be approximated by
x(t0 + ∆t) = x0 + vx0∆t +
1
2
ax0∆t
2
(3)
y(t0 + ∆t) = y0 + vy0∆t +
1
2
ay0∆t
2
(4)
vx(t0 + ∆t) = vx0 + ax0∆t (5)
vy(t0 + ∆t) = vy0 + ay0∆t (6)
1Possibly because you do not have the forcing functions in analytic form or because they are too complicated
to integrate.
1
where the subscript 0 indicates a value at t = t0.
The simulation
We will begin the simulation of the discus flight from when it leaves the thrower’s hand. We will
only consider motion in the x − y plane, and so we ignore any sidewards curving of the discus
flight. A depiction of the simulation is shown in Figure 1. The thrower converts the energy of
her swing into motion of the discus. A large fraction of the energy goes towards translational
energy of the discus. A small fraction goes towards imparting spin on the discus. The spin gives
the discus stability during its flight. We will be concerned with the translational energy as this
gives an initial velocity, v0, of the discus in its flight.
Figure 1: Cartoon of discus flight. The initial flight path angle, γ0, is the angle between the
horizontal and the direction of discus flight as it leaves the hand.
At t = 0, the discus leaves the thrower’s hand. We will assume that the thrower’s hand is
1.8 m above the ground. The discus leaves with a speed of v0 m/s at a flight-path angle, γ0,
with respect to the horizontal. The women’s discus has a mass of 1 kg.
While in flight, we will consider three forces acting on the discus: self-weight due to the
Earth’s gravity (g = 9.81 m/s); aerodynamic lift; and aerodynamic drag. The magnitude of the
lift force is given by
Flift =
1
2
ρv2ArefCL (7)
where ρ = 1.2 kg/m3
is the density of the air, v is the speed of the discus, Aref is a reference
area for the lift coefficient, and CL is a dimensionless lift coefficient that has been determined
by experiment. Similarly, the magnitude of the drag force is
Fdrag =
1
2
ρv2ArefCD (8)
where CD is a dimenionless drag coefficient, also determined by experiment. If we specify all
of our quantities in metres, kilograms and seconds, then the lift and drag forces will be in
Newtons. The direction of the aerodynamic lift force is perpendicular to the direction of the
discus’ velocity, while the aerodynamic drag force is in the opposite direction to that of the
discus’ velocity. The reference area is taken as the area of ths discus when viewed from above.
For a women’s discus, that is Aref = 0.025 m2
.
In practice, the lift and drag coefficients change during flight as the discus changes its angle
of attack. As a simplifying assumption, we will assume that the discus has a constant angle of
attack of 30◦
.
2 This means that we can take the lift and drag coefficients as constant during the
flight. Those constant values are CL = 0.87 and CD = 0.54.
2Note, do not confuse angle of attack with the flight-path angle. In our model of the flight, angle of attack is
taken as constant, but the flight-path angle will vary.
2
Task 3 Draw a free-body diagram showing the forces acting on the discus while it is in flight.
Task 4 Write expressions for the acceleration components ax and ay. Each of these may be built
up as a set of expressions.
Task 5 Write a Python function that computes the trajectory of the discus once it has left the
thrower’s hands and continues until it hits the ground. Your program should start with
the initial state as discussed above and with x0 = 0.0, y0 = 1.8, v0 = 30 m/s and a guessed
value for γ0. It should then integrate the equations of motion with small increments of
time and save the discus’ position and velocity at discrete points in time. Terminate the
integration process when the discus has hit the ground.
Task 6 For CL = 0.0 and CD = 0.0, integrate the equations of motion analytically to determine an
exact solution for the ideal case of no aerodynamic forces. Demonstrate that your program
agrees with your analytic solution.
Task 7 Determine a time-step that gives a sufficiently accurate estimate of the trajectory for the
case of nonzero aerodynamic forces. Use the time of flight and distance of the throw as
the test criteria. Explain why you consider this to be good enough.
Task 8 Use your program to determine the optimum initial flight angle so that the competitor
can beat the world record by the largest possible margin. Assume they are limited to
a throwing speed of 30 m/s. This could be considered a computational experiment and,
although the search for the optimum initial angle could be automated, doing a manual
search is fine for this exercise.
Task 9 Compute and plot the flight path for your optimum initial angle.
Task 10 The men’s discus has a mass of 2 kg, and is larger in size, so its reference area is 0.038 m2
.
Assume the initial throwing speed is 25 m/s for a top male competitor. Use your program
to determine the optimum initial angle for a competitor in the men’s discus event.