# Chapter 12 Kinematics of a Particle¶

## Ex 12.1 Page No 450¶

In [3]:
# Ex 12.1
from scipy import integrate
from scipy.misc import derivative

# Calculation
# Position
t = lambda t: 3*t**2+2*t
s = round(integrate.quad(t, 0, 3)[0],1)  #[meter]
# Acceleration
# a = dv/dt
def f(t):
return 3*t**2+2*t
a = round(derivative(f, 3),1)  #[meter per seconds square]

# Result
print"s = ",(s),"m"
print"a = ",(a),"m/s**(2)"

s =  36.0 m
a =  20.0 m/s**(2)


## Ex 12.2 Page No 451¶

In [5]:
# Ex 12.2
from scipy import integrate
from __future__ import division

# Calculation
# At t = 4 s
v = round(((1/(60**(2))) +0.8*4)**(-0.5),3)  #[meter per second]
# ds = vdt
t = lambda t: ((1/(60**(2))) +0.8*t)**(-0.5)
s = round(integrate.quad(t, 0, 4)[0],2)  #[meter]

# Result
print"v = ",(v),"m/s"
print"s = ",(s),"m"

v =  0.559 m/s
s =  4.43 m


## Ex 12.3 Page No 452¶

In [11]:
# Ex 12.3
from __future__ import division
import math

# Variable Declaration
vA = 75  #[meter per second]
aC = -9.81  #[meter per second square]
sA = 40  #[meter]

# Calculation
# Maximum height
sB = round((-75**(2)/(2*-9.81))+40,1)  #[meter]
# Velocity
# Method 1
vC = round(math.sqrt(0+2*(-9.81)*(0-327)),1)  #[meter per second]
# Method 2
vC = round(math.sqrt(75**(2)+2*(-9.81)*(0-40)),1)  #[meter per second]

# Result
print"sB = ",(sB),"m"
print"vC = ",(vC),"m"

sB =  326.7 m
vC =  80.1 m


## Ex 12.4 Page No 453¶

In [14]:
# Ex 12.4
from scipy import integrate
import math
from __future__ import division

# Calculation
vB = round(2*(0.2**(2)-0.01)**(0.5)*1000,1)  #[millimeter per second]
s = lambda s: (0.5)*(1/math.sqrt(s**(2)-0.01))
t = round(integrate.quad(s, 0.1, 0.2)[0],3)  #[seconds]

# Result
print"vB = ",(vB),"mm/s"
print"t = ",(t),"s"

vB =  346.4 mm/s
t =  0.658 s


## Ex 12.5 Page No 454¶

In [20]:
# Ex 12.5
from __future__ import division

# Calculation
# ds = vdt
# s = t**(3)-3*t**(2) m

# Let s1 be displacement at t = 0 sec
s1 = 0**(3)-3*0**(2)  #[meter]

# Let s2 be displacement at t = 2 sec
s2 = 2**(3)-3*2**(2)  #[meter]

# Let s3 be displacement at t = 0 sec
s3 = 3.5**(3)-3*3.5**(2)  #[meter]

# Let sT be distance travelled in 3.5 sec
sT = round(abs(s2)+abs(s2)+abs(s3),1)  #[meter]

# Velocity
# Let delta_s be displacement from t = 0 to t = 3.5 s
delta_s = s3 - s1  #[meter]

# let v_avg be average velocity
v_avg = delta_s/(3.5-0)  #[meter per second]

# Let vsp_avg be average speed defined in terms of distance travelled sT
vsp_avg = round(sT/(3.5-0),2)  #[meter per second]

# Result
print"sT = ",(sT),"m"
print"v_avg = ",(v_avg),"m/s"
print"vsp_avg = ",(vsp_avg),"m/s"

sT =  14.1 m
v_avg =  1.75 m/s
vsp_avg =  4.03 m/s


## Ex 12.6 Page No 460¶

In [32]:
# Ex 12.6
%matplotlib inline
from matplotlib.pyplot import plot,title,xlabel,ylabel,text,axis,show

# Calculation
# v-t graph
plot(
[0,10,30],
[0,10,10],
color='b',linewidth = 2
)
plot(
[0,10],
[10,10],
color='g',linestyle='--'
)
plot(
[10,10],
[0,10],
color='g',linestyle='--'
)
plot(
[30,30],
[0,10],
color='g',linestyle='--'
)
text(2.6,5.8,'v = t',color='r')
text(15,10.2,'v = 10',color='r')
axis([0,40,0,20])
title('Plot of v-t')
xlabel('t(s)')
ylabel('v(m/s)')
show()
print"\n\n"
# a-t graph
plot(
[0,10],
[1,1],
color='b',linewidth = 2
)
plot(
[10,30],
[0,0],
color='b',linewidth = 2
)
plot(
[10,10],
[0,1],
color='g',linestyle='--'
)
axis([0,40,0,2])
title('Plot of a-t')
xlabel('t(s)')
ylabel('a(m/s**(2))')
show()





## Ex 12.7 Page No 462¶

In [22]:
# Ex 12.7
%matplotlib inline
from matplotlib.pyplot import plot,title,xlabel,ylabel,text,axis,show

# Calculation
# v-t graph
plot(
[0,10,60],
[0,100,0],
color='b',linewidth = 2
)
plot(
[10,10],
[0,100],
color='g',linestyle='--'
)
plot(
[0,10],
[100,100],
color='g',linestyle='--'
)
text(0,95,'v = 10t',color='r')
text(30,66,'v = -2t + 120',color='r')
axis([0,80,0,150])
title('Plot of v-t')
xlabel('t(s)')
ylabel('v(m/s)')
show()
print"\n\n"
# s-t graph
l1 = []
l2 = []
for i in range(0,11,1):
l1.append(i)
l2.append(5*i**(2))
plot(
l1,
l2,
color='b',linewidth = 2
)
l1 = []
l2 = []
for i in range(10,61,1):
l1.append(i)
l2.append(-(i**(2))+120*i-600)
plot(
l1,
l2,
color='b',linewidth = 2
)
plot(
[10,10],
[0,500],
color='g',linestyle='--'
)
plot(
[0,10],
[500,500],
color='g',linestyle='--'
)
plot(
[60,60],
[0,3000],
color='g',linestyle='--'
)
plot(
[0,60],
[3000,3000],
color='g',linestyle='--'
)
text(8,200,'s = 5t**(2)',color='r')
text(25,1500,'s = t**(2) + 120t - 600',color='r')
axis([0,80,0,3500])
title('Plot of s-t')
xlabel('t(s)')
ylabel('s(m)')
show()





## Ex 12.8 Page No 464¶

In [32]:
# Ex 12.8
%matplotlib inline
from matplotlib.pyplot import plot,title,xlabel,ylabel,text,axis,show
from  __future__ import division

# Calculation
# a-s graph
l1 = []
l2 = []
for i in range(0,51,1):
l1.append(i)
l2.append(0.16*i+2)
plot(
l1,
l2,
color='b',linewidth = 2
)
l1 = []
l2 = []
for i in range(50,101,1):
l1.append(i)
l2.append(0)
plot(
l1,
l2,
color='b',linewidth = 2
)
plot(
[50,50],
[0,10],
color='g',linestyle='--'
)
plot(
[0,50],
[10,10],
color='g',linestyle='--'
)
text(4,8,'a = 0.16*s + 2',color='r')
text(52,2,'a = 0',color='r')
axis([0,120,0,15])
title('Plot of s-t')
xlabel('t(s)')
ylabel('s(m)')
show()
print"\n\n"
# Variable Declaration
s = 100  #[meter]
t = s/50 + 10.07  #[seconds]

# Result
print"t = ",(t),"s"



t =  12.07 s


## Ex 12.9 Page No 473¶

In [35]:
# Ex 12.9
import math
from __future__ import division

# Calculation
# Position
x = 16  #[meter]
y = 16**(2)/10  #[meter]
# Let r be straight line distance from A to B
r = round(math.sqrt(x**(2)+y**(2)),1)  #[meter]

# Velocity
vx = 8  #[meter per second]
vy = (2*16*8)/10  #[meter per second]
# Let v be magnitude of velocity at t = 2 s
v = round(math.sqrt(vx**(2)+vy**(2)),1)  #[meter per second]
theta_v = round(math.degrees(math.atan(vy/vx)),1)  #[Degrees]

# Accceleration
ax = 0  #[meter per second square]
ay = 2*8**(2)/10+2*16*0/10  #[meter per second square]
a = round(math.sqrt(ax**(2)+ay**(2)),1)   #[meter per second square]
theta_a = 90  #[Degrees]

# Result
print"r = ",(r),"m"
print"v = ",(v),"m/s"
print"theta_v = ",(theta_v),"degrees"
print"a = ",(a),"m/s**(2)"
print"theta_a = ",(theta_a),"degrees"

r =  30.2 m
v =  26.8 m/s
theta_v =  72.6 degrees
a =  12.8 m/s**(2)
theta_a =  90 degrees


## Ex 12.10 Page No 474¶

In [44]:
# Ex 12.10
import math
from __future__ import division

# Calculation
# Position
r_x = round(0.5*math.sin(1.5),3)  #[meter]
r_y = round(0.5*math.cos(1.5),4)  #[meter]
r_z = round(-0.2*(0.75),4)  #[meter]
r = round(math.sqrt(r_x**(2)+r_y**(2)+r_z**(2)),3)  #[meter]
alpha = round(math.degrees(math.acos(r_x/r)),1)  #[Degrees]
beta = round(math.degrees(math.acos(r_y/r)),1)  #[Degrees]
gamma = round(math.degrees(math.acos(r_z/r)),1)  #[Degrees]

# Velocity
v = round(math.sqrt((1*math.cos(1.5))**(2)+(-1*math.sin(1.5))**(2)+(-0.2)**(2)),3)  #[meter per second]

# Accelaration
a = math.sqrt((-2*math.sin(1.5))**(2)+(-2*math.cos(1.5))**(2))  #[meter per second square]

# Result
print"r_x = ",(r_x),"m"
print"r_y = ",(r_y),"m"
print"r_z = ",(r_z),"m"
print"r = ",(r),"m"
print"alpha = ",(alpha),"degrees"
print"beta = ",(beta),"degrees"
print"gamma = ",(gamma),"degrees"
print"v = ",(v),"m/s"
print"a = ",(a),"m/s**(2)"

r_x =  0.499 m
r_y =  0.0354 m
r_z =  -0.15 m
r =  0.522 m
alpha =  17.1 degrees
beta =  86.1 degrees
gamma =  106.7 degrees
v =  1.02 m/s
a =  2.0 m/s**(2)


## Ex 12.11 Page No 477¶

In [1]:
# Ex 12.11
from __future__ import division
import math

# Variable Declaration
v = 12  #[meters]
# Calculation
# Vertical motion
tAB = round(math.sqrt(-6/((1/2)*(-9.81))),2)  #[seconds]

# Horizontal motion
R = 0+v*1.11  #[meter]

# Result
print"tAB = ",(tAB),"s"
print"R = ",(R),"m"

tAB =  1.11 s
R =  13.32 m


## Ex 12.12 Page No 478¶

In [8]:
# Ex 12.12
import math
from __future__ import division

# Variable Declaration
v0 = 10  #[meter per second]
theta = 30  #[degrees]

# Calculation
vo_x = round(v0*math.cos(math.pi*30/180),2)  #[meter per second]
vo_y = round(v0*math.sin(math.pi*30/180),2)  #[meter per second]

# Horizontal Motion
tOA = round((8-0)/vo_x,4)  #[seconds]

# Vertical Motion
h = round(0+5*tOA+(1/2)*(-9.81)*tOA**(2)+1,3)  #[meter]

# Result
print"h = ",(h),"m"

h =  1.433 m


## Ex 12.13 Page No 479¶

In [13]:
# Ex 12.13
from __future__ import division
import math

# Calculation
# Vertical Motion
vA = round((-1-(1/2)*-9.81*1.5**(2))/(math.sin(math.pi*30/180)*1.5),2)  #[meter per second]

# Horizontal Motion
R = round(0+vA*math.cos(math.pi*30/180)*1.5,1)  #[meter]
h = round(((0**(2)-(vA*math.sin(math.pi*30/180))**(2))/(2*(-9.81)))+1,2)  #[meter]

# Result
print"vA = ",(vA),"m/s"
print"R = ",(R),"m"
print"h = ",(h),"m"

vA =  13.38 m/s
R =  17.4 m
h =  3.28 m


## Ex 12.14 Page No 487¶

In [16]:
# Ex 12.14
import math
from __future__ import division

# Calculation
# Velocity
# Velocity is always directed tangent to the path
vA = 6  #[meter per second]

# Acceleration
rho = round(((1+((1/10)*10)**(2))**(3/2))/(1/10),2)  #[meter]
at = 2  #[meter per second**(2)]
an = round((6**(2))/rho,3)  #[meter per second**(2)]
a = round(math.sqrt(at**(2)+an**(2)),2)  #[meter per second**(2)]
phi = round(math.degrees(math.atan(at/an)),1)  #[Degrees]

# Result
print"vA = ",(vA),"m/s"
print"a = ",(a),"m/s**(2)"

vA =  6 m/s
a =  2.37 m/s**(2)


## Ex 12.15 Page No 488¶

In [17]:
# Ex 12.15
import math
from __future__ import division

# Calculation
# Let t be time needed for acceleration to reach 3 m/s**(2)
t = round(math.sqrt((math.sqrt(3**(2)-2**(2)))/0.04),2)  #[seconds]

# Velocity
# Let v be speed at time t = 7.48 s
v = 3*t  #[meter per second]

# Result
print"t = ",(t),"s"
print"v = ",(v),"m/s"

t =  7.48 s
v =  22.44 m/s


## Ex 12.16 Page No 489¶

In [22]:
# Ex 12.16
import math
from __future__ import division

# Calculation
tB = round((6.142/0.0333)**(1/3),3)  #[seconds]
aBt = 0.2*5.690  #[meter per second square]
vB = 0.1*(tB**(2))  #[meter per second]
aBn = (vB**(2))/2  #[meter per second square]
aB = round(math.sqrt(aBt**(2)+aBn**(2)),2)  #[meter per second square]

# Result
print"aB = ",(aB),"m/s**(2)"

aB =  5.37 m/s**(2)


## Ex 12.18 Page No 498¶

In [25]:
# Ex 12.18
import math
from __future__ import division

# Calculation
vr = 200*1  #[millimeter per second]
vtheta = 100*(1)**(2)*3**(1)  #[millimeter per second]
v = round(math.sqrt(vr**(2)+vtheta**(2)),1)
delta = round(math.degrees(math.atan(vtheta/vr)),1)  #[Degrees]
ar = 200-100*(3**(2))  #[millimeter per second square]
atheta = 100*6+2*200*3  #[millimeter per second square]
a = round(math.sqrt(ar**(2)+atheta**(2)),1)  #[millimeter per second square]
phi = round(math.degrees(math.atan(atheta/ar)),1)   #[Degrees]

# Result
print"v = ",(v),"mm/s"
print"delta = ",(delta),"degrees"
print"a = ",(a),"mm/s**(2)"
print"phi = ",(phi),"degrees"

v =  360.6 mm/s
delta =  56.3 degrees
a =  1931.3 mm/s**(2)
phi =  -68.7 degrees


## Ex 12.19 Page No 499¶

In [4]:
# Ex 12.19
import math
from __future__ import division

# Calculation
vr = round(400*(1/math.cos(math.pi*45/180))*math.tan(math.pi*45/180),1)  #[meter per second]
vtheta = round(100*(1/math.cos(math.pi*45/180))*4,1)  #[meter per second]
v = round(math.sqrt(vr**(2)+vtheta**(2)),1)  #[meter per second]
ar = round((1600*((1/math.cos(math.pi*45/180))*(math.tan(math.pi*45/180)**(2))+(1/math.cos(math.pi*45/180)**(3)))) - 100*(1/math.cos(math.pi*45/180))*4**(2),1)  #[meter per second square]
atheta = round(100*(1/math.cos(math.pi*45/180))*0+2*400*(1/math.cos(math.pi*45/180))*math.tan(math.pi*45/180)*4,1)  #[meter per second square]
a = round(math.sqrt(ar**(2)+atheta**(2)),1)  #[meter per second square]

# Result
print"v = ",(v),"m/s"
print"a = ",(a),"m/s**(2)"


v =  800.0 m/s
a =  6400.0 m/s**(2)


## Ex 12.20 Page No 500¶

In [10]:
# Ex 12.20
import math

# Variable Declaration
a = 10  #[meter per second square]
v = 1  #[meter per second]

# Calculation
theta_dot = math.sqrt((v**(2))-(0**(2)))  #[radian per second]
theta_doubledot = round(math.sqrt((10**(2))-((-1.5)**(2))),2)  #[radian per second square]

# Result
print"theta_doubledot = ",(theta_doubledot),"rad/s**(2)"  # Correction in the answer

theta_dot =  1.0 rad/s
theta_doubledot =  9.89 rad/s**(2)


## Ex 12.21 Page No 506¶

In [11]:
# Ex 12.21

# Calculation
vA = -3*-2  #[meter per second]

# Result
print"vA = ",(vA),"m/s"

vA =  6 m/s


## Ex 12.22 Page No 507¶

In [12]:
# Ex 12.22

# Calculation
vA = -4*-2  #[meter per second]

# Result
print"vA = ",(vA),"m/s"

vA =  8 m/s


## Ex 12.23 Page No 508¶

In [13]:
# Ex 12.23
from __future__ import division

# Calculation
vB = -2/4

# Result
print"vB = ",(vB),"m/s"

vB =  -0.5 m/s


## Ex 12.24 Page No 509¶

In [2]:
# Ex 12.24
from __future__ import division
import math

# Variable Declaration
vA = 0.5  #[meter per second]

# Calculation
vS = round((20*vA)/math.sqrt(225+20**(2)),1)  #[meter per second]
aS = (225*vA**(2))/((225+20**(2))**(3/2))  #[meter per second square]

# Result
print"vS = ",(vS*1000),"mm/s"
print"aS = ",(aS*1000),"mm/s**(2)"

vS =  400.0 mm/s
aS =  3.6 mm/s**(2)


## Ex 12.25 Page No 512¶

In [19]:
# Ex 12.25
import math
from __future__ import division

# Calculation Solution 1 Vector Analysis
vTA_x = round(60-45*math.cos(math.pi*45/180),1)  #[kilometer per hr]
vTA_y = round(-45*math.sin(math.pi*45/180),1)  #[kilometer per hr]
vTA = round(math.sqrt(vTA_x**(2)+vTA_y**(2)),1)  #[kilometer per hr]
theta = round(math.degrees(math.atan(abs(vTA_y)/vTA_x)),1)  #[Degrees]

# Result Solution 1
print"Solution 1"
print"vTA_x  = ",(vTA_x),"km/hr"
print"vTA_y  = ",(vTA_y),"km/hr"
print"vTA  = ",(vTA),"km/hr"
print"theta  = ",(theta),"degrees"

# Calculation Solution 2 Scalar Analysis
# vT = vA + vT/A
vTA_x = round(60-45*math.cos(math.pi*45/180),1)  #[kilometer per hr]
vTA_y = round(-45*math.sin(math.pi*45/180),1)  #[kilometer per hr]

# Result Solution 2
print"Solution 2"
print"vTA_x  = ",(vTA_x),"km/hr"
print"vTA_y  = ",(vTA_y),"km/hr"

Solution 1
vTA_x  =  28.2 km/hr
vTA_y  =  -31.8 km/hr
vTA  =  42.5 km/hr
theta  =  48.4 degrees
Solution 2
vTA_x  =  28.2 km/hr
vTA_y  =  -31.8 km/hr


## Ex 12.26 Page No 513¶

In [21]:
# Ex 12.26
import math
from __future__ import division

# Variable Declaration
rho = 400  #[kilometers]

# Calculation
# Velocity
vBA = 600-700  #[kilometers per hr]
# Acceleration
aBn = 600**(2)/rho  #[kilometers per hr square]
aBA_x = 900  #[kilometers per hr square]
aBA_y = -100-50  #[kilometers per hr square]
aBA = round(math.sqrt(aBA_x**(2)+aBA_y**(2)),1)  #[kilometers per hr square]
theta = round(math.degrees(math.atan(abs(aBA_y)/aBA_x)),1)  #[Degrees]

# Result
print"aBA = ",(aBA),"km/hr**(2)"
print"theta = ",(theta),"degrees"

aBA =  912.4 km/hr**(2)
theta =  9.5 degrees


## Ex 12.27 Page No 514¶

In [27]:
# Ex 12.27
import math
from __future__ import division

# Variable Declaration
vA = 18  #[meter per second]
vB = 12  #[meter per second]

# Calculation
# Velocity
vBA_x = 18*math.cos(math.pi*60/180)  #[meter per second]
vBA_y = 18*math.sin(math.pi*60/180)-12   #[meter per second]
vBA = round(math.sqrt(vBA_x**(2)+vBA_y**(2)),2)  #[meter per second]
theta = round(math.degrees(math.atan(vBA_y/vBA_x)),1)  #[Degrees]

# Acceleration
aBn = vB**(2)/100  #[meter per second square]
aBA_x = round(-1.440-2*math.cos(math.pi*60/180),3)  #[meter per second square]
aBA_y = round(-3-2*math.sin(math.pi*60/180),3)  #[meter per second square]
aBA = round(math.sqrt(aBA_x**(2)+aBA_y**(2)),2)  #[meter per second square]
phi = round(math.degrees(math.atan(aBA_y/aBA_x)),1)  #[Degrees]

# Result
print"vBA = ",(vBA),"m/s"
print"theta = ",(theta),"degrees"
print"aBA = ",(aBA),"m/s**(2)"
print"phi = ",(phi),"degrees"

vBA =  9.69 m/s
theta =  21.7 degrees
aBA =  5.32 m/s**(2)
phi =  62.7 degrees

In [ ]: