
import numpy as np
import pylab as pl
import pandas as pd
import matplotlib.pyplot as plt


M = 49.0 # Mass of particle in kg.
g = 9.8 # Gravitational Acceleration (ms^2).
V = 49  # The initial velocity of the particle (m/s).
α = np.deg2rad(49)  # The initial velocity's angle in degrees.
dr = 0.49  # The coefficient for drag.
dt = 0.0049  # stepsize in s.


# We will prepare the lists to save the variables.
# We will start the velocity and position at t=0 "very important".

t = [0] # The list to keep track of time of the simulation.
vx = [V*np.cos(α)]  # The list for velocity of the x component for the simulation.
vy = [V*np.sin(α)]  # The list for velocity of the y component for the simulation.
x = [0] # The list for x position.
y = [0] # The list for y position.


# The drag force.
drag=dr*V**2 # The drag force. 

# The components of acceleration.
ax = [-(dr*np.cos(α))/M ]          
ay = [-g-(dr*np.sin(α)/M) ]


# We will use the Euler method in this part to revise the 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 
        
    # Corrected velocity
    vx.append(vx[counter]+dt*ax[counter]) # Update the velocity in accordance to Euler's method.
    vy.append(vy[counter]+dt*ay[counter])

    # Corrected position
    x.append(x[counter]+dt*vx[counter])    
    y.append(y[counter]+dt*vy[counter])    

    # Given the indicated new velocity, we can now calculate the drag force and update acceleration.
    vel = np.sqrt(vx[counter+1]**2 + vy[counter+1]**2)  # The magnitude of the particle's velocity.
    drag = dr*vel**2  # The drag force. 
    ax.append(-(drag*np.cos(α))/M)     
    ay.append(-g-(drag*np.sin(α)/M))
    
    # Increment the counter by 1
    counter = counter +1


plt.plot(x,y,'ro')
plt.ylabel("Height (m)")
plt.xlabel("Distance (m)")
    
# The approximate range of the particle should be indicated by the last value of x.

print ("Range of particle is {:3.1f}".format(x[counter]))

print ("Max height {:3.1f}".format(np.max(y)))

Range of particle is 87.8
Max height 35.5


# The initial velocity of the particle (V).
V=49

# The angle of the particle when it was launched (theta).
α=np.deg2rad(49)

# The initial height of the particle (h).
h=49

# The gravitational acceleration (g).
g=9.8


# Component Vx, also known as the horizontal velocity, will equal to.
Vx=V*np.cos(α)

# Component Vy, also known as the vertical velocity, will equal to.
Vy=V*np.sin(α)

#There will be three vectors: V, Vx, and Vy. These three will form a right triangle.

#If the vertical velocity component is equal to 0, then it's the case of horizontal projectile motion.
#If, additionally, α = 90° then it's the case of free fall.

print ("Vy {:3.1f}".format(Vy))
print ("Vx {:3.1f}".format(Vx))

Vy 37.0
Vx 32.1


# The particle's time of flight (t). The time of flight will end when the particle hits the ground. 
#This will happen when the vertical distance from the ground is 0.
t=(V*np.sin(α)+(((V*np.sin(α))**2)+2*g*h)**0.5)/g

print ("Time of flight {:3.1f}".format(t))

Time of flight 8.7


#This is the range of the particle (r).

r=Vx*(Vy+(Vy**2+2*g*h)**0.5)/g

print ("Range of the particle {:3.1f}".format(r))

Range of the particle 279.6


#When the particle reaches the maximum height possible (hmax), it will stop moving up and will then start to fall. 

#It means that its vertical velocity component changes from positive to negative – in other words, 
#it is equal to 0 for a brief moment at time t(Vy=0)
hmax=h+V**2*(np.sin(α))**2/(2*g)

print ("Maximum height possible {:3.1f}".format(hmax))

Maximum height possible 118.8


#Given gravity (g), time (t) and Vy, you should be able to find the initial height.
h_initial=(t**2*g/2)-(t*Vy)

print ("Initial height {:3.1f}".format(h_initial))

Initial height 49.0


# The initial velocity of the particle (V).
V=49

# The initial height of the particle (h).
h=49

# The gravitational acceleration (g).
g=9.8


t=(2*h/g)**0.5 #Time of flight

print ("Time of flight {:3.1f}".format(t))

Time of flight 3.2


r=V*t #Range of the particle

print ("Range of the particle {:3.1f}".format(r))

Range of the particle 155.0


# The gravitational acceleration (g).
g=9.8

# The initial velocity of the particle (V).
V=0

t=4.9 #Time of flight


h=V*t+0.5*g*t**2 #Height

print ("Height {:3.1f}".format(h))

Height 117.6


Vf=V+g*t #Velocity final

print ("Velocity final {:3.1f}".format(Vf))

Velocity final 48.0

Name: Mark Zoren B. Valmonte¶

Group Members: John Eakley, Jessica Vasquez, Kenneth Mosquera-Reinoso¶

Email: valmontem1@montclair.edu¶

Major: Biochemistry¶

Projectile Motion Simulator¶

Introduction¶

Theory¶

Statement of the Problem¶

Projectile Motion with Drag¶

Projectile Motion without Drag¶

Horizontal projectile motion¶

Free Fall¶

Conclusion¶