In [35]:
import numpy as np
import matplotlib.pyplot as plt
% matplotlib inline
# Model parameters
M = 0.008 # Mass of projectile in kg
g = 9.8 # Acceleration due to gravity (m/s^2)
V = 0 # Initial velocity in m/s
ang = 90.0 # Angle of initial velocity in degrees
Cd = 0.005 # Drag coefficient
dt = 0.5 # time step in s
# You can check the variables by printing them out
print V, ang
# Set up the lists to store variables
# Initialize the velocity and position at t=0
t = [0] # list to keep track of time
vx = [V*np.cos(ang/180*np.pi)] # list for velocity x and y components
vy = [V*np.sin(ang/180*np.pi)]
x = [0] # list for x and y position
y = [0.3]
# Drag force
drag = Cd*V**2 # drag force
# Acceleration components
ax = [-drag*np.cos(ang/180*np.pi)/M]
ay = [-g-drag*np.sin(ang/180*np.pi)/M]
## Leave this out for students to try
# We can choose to have better control of the time-step here
dt = 0.001
# Use Euler method to update variables
counter = 0
while (y[counter] >= 0): # Check that the last value of y is >= 0
t.append(t[counter]+dt) # increment by dt and add to the list of time
# Update velocity
vx.append(vx[counter]+dt*ax[counter])
vy.append(vy[counter]+dt*ay[counter])
# Update position
x.append(x[counter]+dt*vx[counter])
y.append(y[counter]+dt*vy[counter])
# With the new velocity calculate the drag force and update acceleration
vel = np.sqrt(vx[counter+1]**2 + vy[counter+1]**2) # magnitude of velocity
drag = Cd*vel**2 # drag force
ax.append(-drag*(vx[counter+1]/vel)/M)
ay.append(-g-drag*(vy[counter+1]/vel)/M)
# Increment the counter by 1
counter = counter +1
# Let's plot the trajectory
plt.plot(t,y,'ro')
plt.ylabel("y (m)")
plt.xlabel("t (s)")
# The last value of x should give the range of the projectile approximately.
print "Range of projectile is {:3.1f} m".format(x[counter])
In [36]:
y1 = y
t1 = t
plt.plot(t1,y1,'ro')
Out[36]:
In [37]:
import numpy as np
import matplotlib.pyplot as plt
% matplotlib inline
# Model parameters
M = 0.008 # Mass of projectile in kg
g = 9.8 # Acceleration due to gravity (m/s^2)
V = 0 # Initial velocity in m/s
ang = 90.0 # Angle of initial velocity in degrees
Cd = 0.1 # Drag coefficient
dt = 0.5 # time step in s
# You can check the variables by printing them out
print V, ang
# Set up the lists to store variables
# Initialize the velocity and position at t=0
t = [0] # list to keep track of time
vx = [V*np.cos(ang/180*np.pi)] # list for velocity x and y components
vy = [V*np.sin(ang/180*np.pi)]
x = [0] # list for x and y position
y = [0.3]
# Drag force
drag = Cd*V**2 # drag force
# Acceleration components
ax = [-drag*np.cos(ang/180*np.pi)/M]
ay = [-g-drag*np.sin(ang/180*np.pi)/M]
## Leave this out for students to try
# We can choose to have better control of the time-step here
dt = 0.001
# Use Euler method to update variables
counter = 0
while (y[counter] >= 0): # Check that the last value of y is >= 0
t.append(t[counter]+dt) # increment by dt and add to the list of time
# Update velocity
vx.append(vx[counter]+dt*ax[counter])
vy.append(vy[counter]+dt*ay[counter])
# Update position
x.append(x[counter]+dt*vx[counter])
y.append(y[counter]+dt*vy[counter])
# With the new velocity calculate the drag force and update acceleration
vel = np.sqrt(vx[counter+1]**2 + vy[counter+1]**2) # magnitude of velocity
drag = Cd*vel**2 # drag force
ax.append(-drag*(vx[counter+1]/vel)/M)
ay.append(-g-drag*(vy[counter+1]/vel)/M)
# Increment the counter by 1
counter = counter +1
# Let's plot the trajectory
plt.plot(t,y,'ro')
plt.ylabel("y (m)")
plt.xlabel("t (s)")
# The last value of x should give the range of the projectile approximately.
print "Range of projectile is {:3.1f} m".format(x[counter])
In [38]:
y2 = y
t2 = t
plt.plot(t2,y2,'ro')
plt.plot(t1,y1,'ro')
Out[38]:
In [39]:
import numpy as np
import matplotlib.pyplot as plt
% matplotlib inline
# Model parameters
M = 0.008 # Mass of projectile in kg
g = 9.8 # Acceleration due to gravity (m/s^2)
V = 0 # Initial velocity in m/s
ang = 90.0 # Angle of initial velocity in degrees
Cd = 0.15 # Drag coefficient
dt = 0.5 # time step in s
# You can check the variables by printing them out
print V, ang
# Set up the lists to store variables
# Initialize the velocity and position at t=0
t = [0] # list to keep track of time
vx = [V*np.cos(ang/180*np.pi)] # list for velocity x and y components
vy = [V*np.sin(ang/180*np.pi)]
x = [0] # list for x and y position
y = [0.3]
# Drag force
drag = Cd*V**2 # drag force
# Acceleration components
ax = [-drag*np.cos(ang/180*np.pi)/M]
ay = [-g-drag*np.sin(ang/180*np.pi)/M]
## Leave this out for students to try
# We can choose to have better control of the time-step here
dt = 0.001
# Use Euler method to update variables
counter = 0
while (y[counter] >= 0): # Check that the last value of y is >= 0
t.append(t[counter]+dt) # increment by dt and add to the list of time
# Update velocity
vx.append(vx[counter]+dt*ax[counter])
vy.append(vy[counter]+dt*ay[counter])
# Update position
x.append(x[counter]+dt*vx[counter])
y.append(y[counter]+dt*vy[counter])
# With the new velocity calculate the drag force and update acceleration
vel = np.sqrt(vx[counter+1]**2 + vy[counter+1]**2) # magnitude of velocity
drag = Cd*vel**2 # drag force
ax.append(-drag*(vx[counter+1]/vel)/M)
ay.append(-g-drag*(vy[counter+1]/vel)/M)
# Increment the counter by 1
counter = counter +1
# Let's plot the trajectory
plt.plot(t,y,'ro')
plt.ylabel("y (m)")
plt.xlabel("t (s)")
# The last value of x should give the range of the projectile approximately.
print "Range of projectile is {:3.1f} m".format(x[counter])
In [40]:
y3 = y
t3 = t
plt.plot(t2,y2,'ro')
plt.plot(t1,y1,'ro')
plt.plot(t3,y3,'ro')
Out[40]:
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