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Total points: 100.

Practicalities about homeworks and projects

  1. You can work in groups (optimal groups are often 2-3 people) or by yourself. If you work as a group you can hand in one answer only if you wish. Remember to write your name(s)!

  2. Homeworks are available approximately ten days before the deadline. You should anticipate this work.

  3. Submission instructions: Submit the paper-and-pencil exercises as a single scanned PDF document. For this homework, this applies to exercises 1–5. Convert the Jupyter notebook to a PDF and attach it to the same PDF document.

Exercise 1 (10 pt), Forces, discussion questions, test your intuition

These questions expect not only an answer, but an explanation of your reasoning.

To receive full credit, these answers should include both the underlying physics that explains your answer and a reflection on your reasoning (for example, whether you are confident in the answer). Physics intuition develops over time and often needs to be reconciled with everyday experience.

Exercise 2 (10 pt), setting up forces, Newton’s second law

Useful material here to read is

  1. Taylor chapters 1.3 and 1.4

  2. Malthe-Sørenssen chapters 5.1, 5.2 and 5.3

A person jumps from an airplane, falling freely for several seconds before the person pulls the cord of the parachute and the parachute unfolds.

Exercise 3 (10 pt), Space shuttle immediately after lift-off

Useful material here to read is: Malthe-Sørenssen chapters 5.1, 5.2 and 5.3

During lift-off of the space shuttle, the engines provide an upward force of 35×106 N35\times 10^{6}\,\mathrm{N}. The mass of the shuttle is approximately 2×106 kg2\times 10^6\,\mathrm{kg}. Take upward as positive. Immediately after lift-off, the shuttle’s speed is approximately zero, so air resistance can be neglected at that instant.

Let us assume that the force from the engines is constant and that the mass of the space shuttle does not change significantly over the first 20 s. Continue to ignore air resistance for this calculation.

Exercise 4 (15 pt), Hitting a golf ball

Useful material here to read is

Do Taylor exercise 1.35. The formulae you obtain here will be useful for the numerical exercises below (see exercise 6 below).

Repeated below consistent with fair use practices

A golf ball is hit from ground level with a speed v0v_0 in a direction due east that is at an angle θ\theta above the horizontal.

Reflect on the form of your answer.

Exercise 5 (15 pt), ball thrown along a sloped ramp

Taylor exercise 1.39. Make sure to draw your setup clearly, show your free-body diagram, and explain any assumptions you make to solve the problem.

Repeated below consistent with fair use practices

A ball is thrown with initial speed v0v_0 up an inclined plane. The plane is inclined at an angle ϕ\phi above the horizontal, and the ball’s initial velocity is at an angle θ\theta above the plane. Choose axes with xx measured up the slope, yy normal to the slope, and zz across it.

R=2v02sin⁡θcos⁡(θ+ϕ)gcos⁡2ϕR=2v_0^2\dfrac{\sin\theta\cos\left(\theta + \phi\right)}{g \cos^2 \phi}
(1)

from its launch point. This is measured up the ramp (i.e., along it).

Rmax=v02g(1+sin⁡ϕ)R_{\text{max}}=\dfrac{v_0^2}{g(1+\sin\phi)}
(2)

Exercise 6 (40pt), Numerical elements, moving to more than one dimension

This exercise should be handed in as a jupyter-notebook through the course submission system. Remember to write your name(s).

Previously, we:

  1. Analytically mapped 1D motion over some time

  2. Gained practice with functions

  3. Reviewed vectors and matrices in Python

In this homework, we will:

  1. Practice using Python syntax and variable manipulation

  2. Utilize analytical solutions to create more refined functions

  3. Work in two, three or even higher dimensions

This material will then serve as background for the numerical part of homework 3. The first part is a simple warm-up, with hints and suggestions you can use for the code to write below.

%matplotlib inline

# As usual, here are some useful packages we will be using. Feel free to use more and experiment as you wish.

import numpy as np
import matplotlib.pyplot as plt
from mpl_toolkits import mplot3d
%matplotlib inline

In class (the falling baseball example) we used an analytical expression for the height of a falling ball. In the first homework we used instead the position from experiment (Usain Bolt’s 100m record run) and stored this information with one-dimensional arrays in Python.

Let us get some practice with this. The cell below creates two arrays, one containing the times to be analyzed and the other containing the xx and yy components of the position vector at each point in time. This is a two-dimensional object. The second array is initially empty. Then we define the initial position to be x=2x=2 and y=1y=1. Take a look at the code and comments to get an understanding of what is happening. Feel free to play around with it.

tf = 4 #length of value to be analyzed
dt = .001 # step sizes
t = np.arange(0.0,tf,dt) # Creates an evenly spaced time array going from 0 to 3.999, with step sizes .001
p = np.zeros((len(t), 2)) # Creates an empty array of [x,y] arrays (our vectors). Array size is same as the one for time.
p[0] = [2.0,1.0] # This sets the initial position to be x = 2 and y = 1

Below we are printing specific values of our array to see what is being stored where. The first number in the array r[]r[] represents which array iteration we are looking at, while the number after the represents which listed number in the array iteration we are getting back.

print(p[0]) # Prints the first array
print(p[0,:]) # Same as above, these commands are interchangeable
[2. 1.]
[2. 1.]

print(p[3999]) # Prints the 4000th array
[0. 0.]

print(p[0,0]) # Prints the first value of the first array
2.0

print(p[0,1]) # Prints the second value of first array
print(p[:,0]) # Prints the first value of all the arrays
1.0
[2. 0. 0. ... 0. 0. 0.]

Then try running this cell. Notice how it gives an error since we did not implement a third dimension into our arrays

#print(p[:,2])

In the cell below we want to manipulate the arrays. In this example we make each vector’s xx component valued the same as their respective vector’s position in the iteration and the yy value will be twice that value, except for the first vector, which we have already set. That is we have p[0]=[2,1],p[1]=[1,2],p[2]=[2,4],p[3]=[3,6],...p[0] = [2,1], p[1] = [1,2], p[2] = [2,4], p[3] = [3,6], ...

Here we set up an array for xx and yy values.

for i in range(1, len(p)):
    p[i] = [i,2*i]
# Checker cell to make sure your code is performing correctly
c = 0
for i in range(len(p)):
    if i == 0:
        if p[i,0] != 2.0:
            c += 1
        if p[i,1] != 1.0:
            c += 1
    else:
        if p[i,0] != 1.0*i:
            c += 1
        if p[i,1] != 2.0*i:
            c += 1

if c == 0:
    print("Success!")
else:
    print("There is an error in your code")
Success!

You could also think of an alternative way of storing the above information. Feel free to explore how to store multidimensional objects.

In the previous homework we studied Usain Bolt’s 100m run and in class we studied a falling baseball. We made basic plots of the baseball moving in one dimension. Now we will be working with a three-dimensional variant. This will be useful for our next homeworks and numerical projects.

Assume we have a soccer ball moving in three dimensions with the following trajectory:

x(t)=10tcos⁡45∘x(t) = 10t\cos{45^{\circ}}
(3)
y(t)=10tsin⁡45∘y(t) = 10t\sin{45^{\circ}}
(4)
z(t)=10t−9.812t2z(t) = 10t-\dfrac{9.81}{2}t^2
(5)

Now let us create a three-dimensional (3D) plot using these equations. In the cell below we write the equations into their respective labels. We fix a final time in the code below.

Important Concept: Numpy comes with many mathematical packages, some of them being the trigonometric functions sine, cosine, tangent. We are going to utilize these here. Additionally, these functions work with radians, so we will also be using a function from Numpy that converts degrees to radians.

tf = 2.04  # The final time to be evaluated
dt = 0.1  # The time step size
t = np.arange(0,tf,dt) # The time array
theta_deg = 45 # Degrees
theta_rad = np.radians(theta_deg) # Converts degrees to their radian counterparts
x = 10*t*np.cos(theta_rad) # Equation for our x component, utilizing np.cos() and our calculated radians
y = 10*t*np.sin(theta_rad) # Put the y equation here
z = 10*t-9.81/2*t**2# Put the z equation here

Then we plot it

## Once you have entered the proper equations in the cell above, run this cell to plot in 3D
fig = plt.axes(projection='3d')
fig.set_xlabel('x')
fig.set_ylabel('y')
fig.set_zlabel('z')
fig.scatter(x,y,z)
<mpl_toolkits.mplot3d.art3d.Path3DCollection at 0x7f24b729b740>
<Figure size 640x480 with 1 Axes>

Define the position vector as r = np.array([x, y, z]), then run the code and plot using the array rr.

## Run this code to plot using our r array 
# fig = plt.axes(projection='3d')
# fig.set_xlabel('x')
# fig.set_ylabel('y')
# fig.set_zlabel('z')
r = np.array([x, y, z])
fig.scatter(r[0],r[1],r[2])
<mpl_toolkits.mplot3d.art3d.Path3DCollection at 0x7f24b4fd9430>

Complete Exercise 4 above (Taylor exercise 1.35) before moving further. (Recall that the golf ball was hit due east at an angle θ\theta with respect to the horizontal, and the coordinate directions are xx measured east, yy north, and zz vertically up.)

Homework
Homework 1
Homework
Homework 3