In [1]:

```
import math
# Variables:
u1 = 0
v1 = 72.*1000./3600 #m/s
s1 = 500. #m
# Solution:
# Calculating the initial acceleration of the car
a1 = (v1**2-u1**2)/(2*s1) #m/s**2
#Calculating time taken by the car to attain the speed
t1 = (v1-u1)/a1 #seconds
#Parameters for the second case
u2 = v1
v2 = 90.*1000/3600 #m/s
t2 = 10. #seconds
#Calculating the acceleration for the second case
a2 = (v2-u2)/t2 #m/s**2
#Calculating the distance moved by the car in the second case
s2 = (u2*t2)+(a2/2*t2**2)
#Parameters for the third case
u3 = v2
v3 = 0 #m/s
t3 = 5 #seconds
#Calculating the distance moved by the car
s3 = (u3+v3)*t3/2 #m
#Results:
print " The acceleration of the car, a = %.1f m/s**2. "%(a1)
print " The car takes t = %d s to attain the speed."%(t1)
print " The acceleration of the car in the second case, a = %.1f m/s**2."%(a2)
print " The distance moved by the cars = %d m."%(s2)
print " The distance travelled by the car during braking, s = %.1f m."%(s3)
```

In [1]:

```
# variables
t = 1. # second
v = 6.25 # m/s
# calculations and results
C1 = v - 0.25 +t -5
# when t = 2
t = 2.
v = t**4/4 - t**3 + 5*t + 2
print "Velocity at t=2 seconds, V = %.f m/s"%v
# when t = 1 seconds and s = 8.30 m.
t = 1.
s = 8.30
C2 = s - 1./20 + 1./4 - 5./2 - 2
t = 2. # seconds
s = t**5/20 - t**4/4 + 5*t**2/2 + 2*t + 4
print "Displacement = %.1f m"%s
```

In [2]:

```
import math
from scipy.integrate import quad
# Variables:
#Initial parameters
v0 = 100. #kmph
t0 = 0
#Parameters at the end of 40 seconds
v1 = 90./100*v0 #kmph
t1 = 40. #seconds
#Solution:
#The acceleration is given by
#a = (-dv/dt) = k*v
#Integrating
#we get ln(v) = -k*t+C
#Calculating the constant of integration
def f3(v):
return 1./v
C = quad(f3,1,100)[0]
#Calculating the constant of proportionality
k = (C-2.3*math.log10(90))/40
#Time after 120 seconds
t2 = 120. #seconds
#Calculating the velocity after 120 seconds
v120 = 10**((-k*t2+C)/2.29)
#Results:
print " The velocity at the end of 120 seconds = %.1f kmph."%(v120)
```

In [3]:

```
%matplotlib inline
import math
from matplotlib.pyplot import *
# Variables:
s = 500. #mm
s1 = 125. #mm
s2 = 250. #mm
s3 = 125. #mm
t = 1. #second
#Solution:
#Matrices for the velocity vs. time graph
V = [0 ,750.,750.,0] #The velocity matrix
T = [0,1./3,2./3,1] #The time matrix
plot(T,V)
xlabel("Time")
ylabel("Velocity")
#Calculating the time of uniform acceleration
#Equating the time taken to complete the stroke to 1 second
v = (125/(1./2)+250/1+125/(1./2))/1 #mm/s
#Results:
show()
print " The maximum cutting speed v = %d mm/s."%(v)
```

In [7]:

```
# Variables:
N0 = 0
N = 2000. #rpm
t = 20. #seconds
#Solution:
#Calculating the angular velocities
omega0 = 0
omega = 2*math.pi*N/60 #rad/s
#Calculating the angular acceleration
alpha = (omega-omega0)/t #rad/s**2
#Calculating the angular distance moved by the wheel during 2000 rpm
theta = (omega0+omega)*t/2 #rad
#Calculating the number of revolutions made by the wheel
n = theta/(2*math.pi)
#Results:
print " The angular acceleration of the wheel, alpha = %.3f rad/s**2."%(alpha)
print " The wheel makes n = %.1f revolutions."%(n)
```

In [4]:

```
import math
# Variables:
r = 1.5 #m
N0 = 1200.
N = 1500. #rpm
t = 5. #seconds
#Solution:
#Calculating the angular velocities
omega0 = 2*math.pi*N0/60
omega = 2*math.pi*N/60 #rad/s
#Calculating the linear velocity at the beginning
v0 = r*omega0 #m/s
#Calculating the linear velocity at the end of 5 seconds
v5 = r*omega #m/s
#Calculating the angular acceleration
alpha = (omega-omega0)/t #ad/s**2
#Calculating the math.tangential acceleration after 5 seconds
TangentialAcceleration = alpha*(r/2) #m/s**2
#Calculating the radial acceleration after 5 seconds
RadialAcceleration = (round(omega)**2)*(r/2) #m/s**2
#Results:
print " The linear velocity at the beginning, v0 = %.1f m/s."%(v0)
print " The linear velocity after 5 seconds, v5 = %.1f m/s."%(v5)
print " The tangential acceleration after 5 seconds is %.1f m/s**2."%(TangentialAcceleration)
print " The radial acceleration after 5 seconds is %.f m/s**2."%(RadialAcceleration)
```