The first exercise is meant to remind ourselves about properties of the exponential function and imaginary numbers. This is highly relevant later in this course when we start analyzing oscillatory motion and some wave mechanics. As physicists we should thus feel comfortable with expressions that include \( \exp{(\imath\omega t)} \). Here \( t \) could be interpreted as time and \( \omega \) as a frequency and \( \imath \) is the imaginary unit number.
Exercise 1.18 in the textbook of Taylor, Classical Mechanics. Part (1.18a) gives 5pt and part (1.18b) gives also 5pt.
This exercise should be handed in as a jupyter-notebook at D2L. Remember to write your name(s).
Our first numerical attempt will involve reading data from file or just setting up two vectors, one for position and one for time. Our data are from Usain Bolt's world record 100m during the olympic games in Beijing in 2008. The data show the time used in units of 10m (see below). Before we however venture into this, we need to repeat some basic Python syntax with an emphasis on
For more information, see the introductory slides. Here are some of the basic packages we will be using this week
import numpy as np
import matplotlib.pyplot as plt
%matplotlib inline
The first exercise here deals with simply getting familiar with vectors and matrices.
We will be working with vectors and matrices to get you familiar with them.
V1 = np.array([1,2,3])
V2 = np.array([4,5,6])
print("V1: ", V1)
print("V2: ", V2)
If this is not too familiar, here's a useful link for creating vectors in python https://docs.scipy.org/doc/numpy-1.13.0/user/basics.creation.html. Alternatively, look up the introductory slides.
Now let us do some basic mathematics with vectors.
Compute and print the following, and double check with your own paper and pencil calculations:
Here is some useful explanation on numpy array operations if you feel a bit confused by what is happening, see https://www.pluralsight.com/guides/overview-basic-numpy-operations.
The following code prints the first two exercises
print(V1-V2)
print(V2-V1)
For the dot product of V1 and V2 below we can use the dot function of numpy as follows
print(V1.dot(V2))
As a small challenge try to write your own function for the dot product of two vectors.
Matrices can be created in a similar fashion in python. In this language we can work with them through the package numpy (which we have already imported)
M1 = np.matrix([[1,2,3],
[4,5,6],
[7,8,9]])
M2 = np.matrix([[1,2],
[3,4],
[5,6]])
M3 = np.matrix([[9,8,7],
[4,5,6],
[7,6,9]])
Matrices can be added in the same way vectors are added in python as shown here
print("M1+M3: ", M1+M3)
What happens if we try to do \( M1+M2 \)?
That's enough vectors and matrices for now. Let's move on to some physics problems!
We can opt for two different ways of handling the data. The data are listed in the table here and represents the total time Usain Bolt used in steps of 10 meters of distance. The label \( i \) is just a counter and we start from zero since Python arrays are by default set from zero. The variable \( t \) is time in seconds and \( x \) is the position in meters. You may find it useful to include results at time \( t=0 \) as well. The position is obviously then \( x=0 \) m.
| i | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
| x[m] | 10 | 20 | 30 | 40 | 50 | 60 | 70 | 80 | 90 | 100 |
| t[s] | 1.85 | 2.87 | 3.78 | 4.65 | 5.50 | 6.32 | 7.14 | 7.96 | 8.79 | 9.69 |
The following example code may help here
# we just initialize time and position
x = np.array([10.0, 20.0, 30.0, 40.0, 50.0, 60.0, 70.0, 80.0, 90.0, 100.0])
t = np.array([1.85, 2.87, 3.78, 4.65, 5.50, 6.32, 7.14, 7.96, 8.79, 9.69])
plt.plot(t,x, color='black')
plt.xlabel("Time t[s]")
plt.ylabel("Position x[m]")
plt.title("Usain Bolt's world record run")
plt.show()