# Chapter 12:Kinematics of Linear Motion¶

## Example 12.1,Page No.437¶

In [1]:
import math

#Initilization of Variables

u=2 #m/s #Initial velocity
v=5 #m/s #Final Velocity
t=4 #sec

#Calculations

a=(v-u)*t**-1 #m/s**2

#Result
print"Acceleration of the body is",round(a,2),"m/s**2"

Acceleration of the body is 0.75 m/s**2


## Example 12.2,Page No.437¶

In [2]:
import math

#Initilization of Variables

u=15 #m/s #Intital velocity
v=0 #m/s #Final velocity
t=5 #sec

#Calculations

#acceleration
a=u*t**-1 #m/s**2

#Distance
S=u*t-a*t**2*0.5

#Result
print"Retardation is",round(a,2),"m/s**2"
print"Distance travelled by the car is",round(S,2),"m"

Retardation is 3.0 m/s**2
Distance travelled by the car is 37.5 m


## Example 12.3,Page No.437¶

In [3]:
import math

#Initilization of Variables

u=250 #m/s #Initial Velocity
v=0  #m/s #final Velocity
s1=0.40 #m #Distance
s2=0.20 #m #distance moved

#Calculations

#acceleration
a=u**2*(2*s1)**-1 #m/s**2

#velocity
v=(u**2-2*a*s2)**0.5 #m/s

#Result
print"Acceleration is",round(a,2),"m/s**2"
print"Velocity is",round(v,2),"m/s"

Acceleration is 78125.0 m/s**2
Velocity is 176.78 m/s


## Example 12.4,Page No.438¶

In [4]:
import math

#Initilization of Variables

L_AB=100 #m #Distance AB
L_BC=100 #m #distance BC
t1=10 #s #Time taken by car from A to B
t2=8 #s #Time taken by car from B to C
s=100 #m #Distance
L_AC=L_AB+L_BC #m

#Calculations

#From equation of distance we get value of velocity
#s=10*V_A+50*a   ...........................1

#again using equation of distance we get
#L_AC=18*V_A+162*a   .........................2

#After simplifying above equations we get equations like
#900=90*V_A+450*a   ......................3
#1000=90*V_A+810*a   ..............4

#Subtracting equations 3 by 4 we get
a=100*360**-1 #m/s**2

#Velocity of car A
V_A=(100-13.9)*10**-1 #m/s

#Velocity of car B
V_B=(L_BC-(a*t2**2*0.5))*t2**-1 #m/s

#Distance OA
s3=V_A**2*(2*a)**-1 #m/s

#Result
print"Acceleration of car is",round(a,2),"m/s**2"
print"Velocity of car A",round(V_A,2),"m/s"
print"Velocity of car B",round(V_B,2),"m/s"
print"distance of mark A from starting point is",round(s3,2),"m"

Acceleration of car is 0.28 m/s**2
Velocity of car A 8.61 m/s
Velocity of car B 11.39 m/s
distance of mark A from starting point is 133.44 m


## Example 12.5,Page No.440¶

In [5]:
import math

#Initilization of Variables

u=0 #m/s  #Initial velocity
a=2 #m/s**2 #Acceleration
u2=40 #m/s #Uniform velocity

#Calculations

#after simplifying Distance travelled we get
#t**2+20t+100=0  .................1

#Distance travelled by police party=40*t ..........2

#Equating equations 1 and 2 we get
#t**2-20*t+100
a=1
b=-20
c=100

X=b**-4*a*c

t=(-b+X**0.5)*(2*a)**-1 #s

#Result
print"TIme taken in which police van will overtake the car is",round(t,2),"s"

TIme taken in which police van will overtake the car is 10.01 s


## Example 12.6,Page No.440¶

In [6]:
import math

#Initilization of Variables

v=10 #m/s #uniform Velocity
a=1 #m/s**2 #Uniform acceleration
u=10 #m/s #uniform velocity

#Calculations

#From distance equation and further simplifying we get
#S=(t**2+100-20*t)  .................1

#again sub value in distance we get
#S=u*t   ............2

#Equating two equations and further simplifying we get
#t**2-40*t+100=0
a=1
b=-40
c=100

X=b**2-4*a*c

t1=(-b+X**0.5)*(2*a)**-1 #s
t2=(-b-X**0.5)*(2*a)**-1 #s

#time required to catch smugglers car is
t3=(t1-10)

#Result
print"Time necesscary for the jeep to catch up with the smuggler's car is",round(t3,2),"s"

Time necesscary for the jeep to catch up with the smuggler's car is 27.32 s


## Example 12.6(A),Page No.442¶

In [7]:
import math

#Initilization of Variables

u=0
a=4 #m/s**2
t1=7 #s
t2=6 #s

#Calculations

#Distance travelled in 7 seconds
S7=u*t1+0.5*a*t1**2 #m

#DistANCE TRAVELLED in 6 seconds
S6=u*t2+0.5*a*t2**2 #m

#Distance travelled in 7th second
S7_2=S7-S6

#Result
print"Distance travelled in 7th second is",round(S7_2,2),"s"

Distance travelled in 7th second is 26.0 s


## Example 12.7,Page No.442¶

In [8]:
import math

#Initilization of Variables

S5=15 #m  #Distance travelled for 5 th seconds
S10=25 #m #Distance travelled for 10 th seconds
n1=10
n2=5

#Calculations

#Equation for distance covered for nth seconds
#S=u+a*2**-1

#distance covered in 10 th second
#S10=u+a*2**-1*(2*n1-1)  ...................1

#distance covered in 5 th second
#S5=u+a*2**-1*(2*n2-1)  .........................2

#Subtracting equation 2 by 1 we get
a=10*(19-9)**-1*2

u=S5-9*2**-1*2

#Result
print"Initial Velocity of the body is",round(a,2),"m/s**2"
print"Acceleration of the body is",round(u,2),"m/s**2"

Initial Velocity of the body is 2.0 m/s**2
Acceleration of the body is 6.0 m/s**2


## Example 12.8,Page No.443¶

In [9]:
import math

#Initilization of Variables

h=90 #3mm Height of tower
h1=30 #m #Height at which both particles meet
S1=60 #m #Distance travelled by first particle
S2=30 #m
g2=-9.81 #m/s**2

#For Initial Velocity
u1=0
g=9.81 #m/s**2

#Calculations

#Time
t1=((S1*2)*g**-1)**0.5 #s

#For second particle
u2=(S2-0.5*g2*t1**2)*t1**-1

#Result
print"Velocity with which second particle projected is",round(u2,2),"m/s"

Velocity with which second particle projected is 25.73 m/s


## Example 12.9,Page No.443¶

In [10]:
import math

#Initilization of Variables

h=800 #m #Height of aeroplane
U=166.67 #m/s

#First case
u1=0
h2=800 #m #Height of bomb when released
g=9.81 #m/s**2

#Calculations

#Time required to reach the ground
t=(h*2*(g)**-1)**0.5

#Horizontal distance travelled
S=U*t

#Result
print"Time required to reach the ground is",round(t,2),"s"
print"Horizontal distance travelled is",round(S,2),"m"

Time required to reach the ground is 12.77 s
Horizontal distance travelled is 2128.55 m


## Example 12.10,Page No.445¶

In [11]:
import math

#Initilization of Variables

u=0 #m/s #Initial velocity
g=9.81 #m/s**2 #acceleration due to gravity

#Calculations

#t1+t2=4   ...........1

#Depth of well after simplifying we get
#h=4.905*t1**2   ........2

#Time taken by sound to reach from bottom of well
#t2=4.905*t1**2*350**-1 #s   ...............3

#Sub value in equation 1 and further simplifying we get
#4.905*t1**2+350*t1-1400=0
a=4.905
b=350
c=-1400

X=b**2-4*a*c

t1=(-b+X**0.5)*(2*a)**-1 #s

#Depth of well
h=4.905*t1**2 #m

#Result
print"Depth of well is",round(h,2),"m"

Depth of well is 70.75 m


## Example 12.11,Page No.446¶

In [12]:
import math

#Initilization of Variables

u=19.6 #m/s #Initial velocity
h=24.5 #m #height of tower
g=9.80 #m/s**2 #acceleration de to gravity

#Calculations

#Max height of stone
h1=(u**2)*(2*g)**-1 #m

#Time for stone to move from A to C
t1=u*g**-1

#Time for stone to move from C to D
h2=h+h1 #m #Max height to which stone will rise
t2=((h2*4.9**-1))**0.5 #s

#Total time for stone to reach the ground
t=t1+t2 #s

#Result
print"Total time for stone to reach the ground is",round(t,2),"s"
print"Velocity of stone in downward travel is",round(u,2),"m/s"
print"Max height to which the stone will rise is",round(h2,2),"m"

Total time for stone to reach the ground is 5.0 s
Velocity of stone in downward travel is 19.6 m/s
Max height to which the stone will rise is 44.1 m


## Example 12.12,Page No.447¶

In [13]:
import math

#Initilization of Variables

u=0 #Initial Velocity
t=5 #s #time taken by stone in striking the glass pane
g=9.81 #m/s**2

#Calculations

#velocity
v1=u+g*t #m/s

#Velocity lost in breaking stones
v2=20*100**-1*10

#Velocity of the stone after breaking the glass pane
v3=v1-v2 #m/s

#distance travelled in t2=1 s
t2=1 #s
s=v3*t2+0.5*g*t2**2 #m

#Result
print"Distance travelled by the stone in next second is",round(s,2),"m"

Distance travelled by the stone in next second is 51.96 m


## Example 12.13,Page No.448¶

In [14]:
import math

#Initilization of Variables

u=0 #m/s
s=53.90 #m #Distance
g=9.80 #m/s**2 #Acceleration due to gravity

#Calculations

#Height
#After simplifying equation of distance we get
#h1=4.9*t**2  ..................................1

#Distance travelleed in (t-1) s
#After simplifying equation of distance we get
#h2=4.9(t-1)**2   ..................................2

#Distance travelled by object in last seconds
#h3=h-h2
#After substituting values in above equation we get
#h3=4.9(2*t-1)

#Equating h3 to s we get after simplifying
t=12*2**-1 #m/s

#height from which object falls
h1=4.9*t**2

#Result
print"height from which object falls is",round(t,2),"s"
print"Total time taken by object in falling is",round(h1,2),"m"

height from which object falls is 6.0 s
Total time taken by object in falling is 176.4 m


## Example 12.14,Page No.448¶

In [15]:
import math

#Initilization of Variables

g=9.80 #m/s**2
u=0 #m/s

#Calculations

#Distance travelled in time t after simplifiying
#h=4.9*t**2

#Distance travelled in (t-1)s
#h-h2=2*3**-1*h

#Substituting value we get equation as
#2*t**2-6*t+3=0
a=2
b=-6
c=3

X=b**2-4*a*c

t=(-b+X**0.5)*(2*a)**-1 #s
t2=(-b-X**0.5)*(2*a)**-1 #s

#Height of tower
h=4.9*t**2

#Result
print"Height of tower is",round(h,2),"m"

Height of tower is 27.43 m


## Example 12.15,Page No.449¶

In [16]:
import math

#Initilization of Variables

u1=30 #m/s #Initial Vlocity of 1st object
u2=40 #m/s #Initial Velocity of2nd object

#Calculations

#For the first object
#After simplifying we get
#h1=30*t-4.905*t**2    ................1

#For second object
##After simplifying we get
#h2=40*(t-4)-4.905(t-4)**2    ...........2

#Equating equations 1 and 2 and further simplify we get
t=238.48*49.24**-1 #s

#height
h=30*t-4.905*t**2 #m

#Result
print"Time whrn the two objects will meet each other is",round(t,2),"s"
print"Height from the earth at which the two objects will meet is",round(h,2),"m"

Time whrn the two objects will meet each other is 4.84 s
Height from the earth at which the two objects will meet is 30.24 m


## Example 12.16,Page No.450¶

In [17]:
import math

#Initilization of Variables

h=100 #m #Height of tower
u1=0 #Initial velocity of 1st particle
S2=30 #m #Distace travelled by 2nd particle
S1=70 #m #Distance travelled by 1st paerticle
g=9.81 #m/s**2

#Calculations

#time of particle 1
t=(S1*(g*2**-1)**-1)**0.5 #s

#Initial velocity
u=((S2+(g*2**-1)*t**2)*t**-1) #m/s

#Result
print"Velocity with which the second particle is projected upward is",round(u,2),"m/s"

Velocity with which the second particle is projected upward is 26.47 m/s


## Example 12.16(A),Page No.451¶

In [18]:
import math

#Initilization of Variables

n=5 #Rate of drops
u=0 #Initial Velocity
v=3 #m/s #Final Velocity
g=9.81 #m/s**2
t=3*9.81**-1 #s

#Calculations

#Vertical Distance
Sb=u*t+0.5*g*t**2 #m

#time taken by drop A
t2=3*9.81**-1-0.2 #s

#Vertical Distance travelled from mouth of faucet by drop A
Sa=u*t2+0.5*g*t2**2 #m

#Vertical Separation between drops A and B
S=Sb-Sa #m

#Result
print"Vertical separation between two drops is",round(S,3),"m"

Vertical separation between two drops is 0.404 m


## Example 12.18,Page No.453¶

In [19]:
import math

#Initilization of Variables

u=0 #m/s**2 #Initial Velocity
#S=(x+20) #m #distance
g=9.81 #m/s**2

#Calculations

#Distance travelled by Body in time t
#x=0.5*g*t**2   .................................1

#Distance travelled by body in time (t+0.4)s
#S2=0.5(t**2+0.**t+0.16)  ........................2

#Subtracting equation 2 from 1 we get
t=3.92*0.80**-1

#Distance travelled by Body in time t is given by
x=0.5*g*t**2 #m

#Result
print"Distance travelled by Body in time t is",round(x,2),"m"

Distance travelled by Body in time t is 117.77 m


## Example 12.19,Page No.454¶

In [20]:
import math

#Initilization of Variables

#Equation of displacement
#s=t**3+3*t**2+4*t+5

#Calculations

#After differentiating displacement equation we get velocity at t=0
#At
t=0
v=3*t**2+6*t+4 #m/s

#Velocity at t=4 seconds
t2=4
v2=3*t2**2+6*t2+4 #m/s

#After differentiating w.r.to t we get equation of acceleration as
#a=6*t+6

#at t=0
t3=0
a=6*t3+6 #m/s**2

#at t=4
t4=4
a2=6*t4+6 #m/s**2

#Result
print"Velocity at start of 4seconds is",round(v,2),"m/s"
print"Velocity after 4seconds is",round(v2,2),"m/s"
print"Acceleration at start is",round(a,2),"m/s**2"
print"Acceleration after four seconds is",round(a2,2),"m/s**2"

Velocity at start of 4seconds is 4.0 m/s
Velocity after 4seconds is 76.0 m/s
Acceleration at start is 6.0 m/s**2
Acceleration after four seconds is 30.0 m/s**2


## Example 12.20,Page No.455¶

In [21]:
import math

#Initilization of Variables

#Equation of particle motion
#s=18*t+3*t**2-2*t**3

#Calculations

#After differentiating w.r.to t to above equation we get
#6*t**2-6*t-18

#at
t=0
#Velocity
v=6*t**2+6*t+18

#After differentiating above equation again we get equation of acceleration
#at
t2=0
a=6-12*t2

#After differentiating equation of velocity we get value of
t2=6*12**-1

vmax=18+6*t2-6*t2**2 #m/s

#Result
print"Velocity at start is",round(v,2),"m/s"
print"Acceleration at start is",round(a,2),"m/s**2"
print"Time when it reaches max velocity is",round(t2,2),"s"
print"MAx velocity of particle is",round(vmax,2),"m/s"

Velocity at start is 18.0 m/s
Acceleration at start is 6.0 m/s**2
Time when it reaches max velocity is 0.5 s
MAx velocity of particle is 19.5 m/s


## Example 12.21,Page No.457¶

In [22]:
import math

#Initilization of Variables

#a=-**s**-2 #m/s**2
t=1 #s #time
s=4 #m #Distance
v=2 #m/s #Velocity
t2=2 #s #time

#Calculations

#Acceleration equation
#a=v*(dv*ds**-1)
#After sub values and further simplifying and integrating the obtained equation we get
#v**2=8*s**-1+C1   ...................1
#Sub equation in above equations we get
C1=v**2*2**-1-8*s**-1

#Sub value in equation 1 and furter simplifying and integrating obtained equation we get
#2*3**-1*s**(3*2**_1)=4*t+c2
#Sub values
C2=2*3**-1*s**(3*2**-1)-4*t

#Sub value of C2 and further sub values we get
s=14**(2*3**-1)

#Acceleration
a=8*s**-2 #m/s**2

#Result
print"Acceleration when t=2 is",round(a,4),"m/s**2"

Acceleration when t=2 is 0.2371 m/s**2


## Example 12.22,Page No.458¶

In [23]:
import math

#Initilization of Variables

#a=4*t**2-2
t1=0
s1=-2 #m

t2=2 #s
s2=-20 #m

t3=4 #s

#Calculations

#a=4*t**2-2
#After integrating above equations we getand further simplifying we get equation of distance as
#s=t**4*3**-1-t**2+C1*t+C2
C2=s1-t1**4*3**-1

#s=t**4*3**-1-t**2+C1*t-2   ...............3
#Now after sub in equation and further simplifying the equation we get
C1=(s2-t2**4*3**-1+t2**2+2)*t2**-1
#Sub in above equation 3 we get

#when t=4
t4=4
s3=t4**4*3**-1-t4**2+C1*t4-2

#Result
print"Position of particle when t=4 s is",round(s3,2),"s"

Position of particle when t=4 s is 28.67 s


## Example 12.23,Page No.460¶

In [24]:
import math

#Initilization of Variables

#v=2*t**3-t**2-2*t**2-2*t+4
t=2 #s
s=10 #m
t2=6 #s

#Calculations

#acceleration
a=6*t2**2-2*t2-2 #m/s**2

#Displacement when t=6s
#After integrating and further simplifying the equation of velocity we get equation of displacement as
#s=2*t**4*4**-1-t**3*3**-1-t**2+4*t+C   ...............1
#After sub values we get
C=s-(2*t**4*4**-1-t**3*3**-1-t**2+4*t)

#Sub value of C in equation
s2=2*t2**4*4**-1-t2**3*3**-1-t2**2+4*t2+C #m

#Result
print"Acceleration is",round(a,2),"m/s**2"
print"Displacement of particle when t=6 s is",round(s2,2),"m"

Acceleration is 202.0 m/s**2
Displacement of particle when t=6 s is 564.67 m


## Example 12.24,Page No.461¶

In [57]:
import math

#Initilization of Variables

#a=2-3*t
t=5 #s
t2=10 #s
v=20 #m/s #velocity
s=85 #m  #displacement

#Calculations

#Acceleration at start when t=0
a=2-3*t

#Equation of acceleration after integrating and further simplifying we get
C1=v-(2*t-3*t**2*2**-1)

#After substituting in equation and further simplifying we get
#At t3=0
t3=0
v=2*t3-3*t3**2*2**-1+C1

#After differentiating equation of velocity and integrating it we get equation of displacement as
#s=t**2-3*t**3*6**-1+47.5*t+C2
C2=s-(t2**2-3*t2**3*6**-1+47.5*t2)
#sub value of C2 in above equation we get
s2=t3**2-3*t3**3*6**-1+47.5*t3+C2

#Sub v=0 in equation 3 we get an duaqratic equation as
#3*t**2-4*t-95=0
a=3
b=-4
c=-95

X=b**2-4*a*c

t4=(-b+(X**0.5))*(2*a)**-1 #s
#Sub value of t4 in equation of displacement and we get
s3=t4**2-3*t4**3*6**-1+47.5*t4+C2

#Result
print"Acceleration from origin at start of observation is",round(a,2),"m/s**2"
print"Velocity from origin at start of observation is",round(v,2),"m/s**2"
print"distance from origin at start of observationis",round(s2,2),"m"
print"Time after start of observation is",round(t4,2),"s"
print"Distance from origin is",round(s3,2),"m"

Acceleration from origin at start of observation is 3.0 m/s**2
Velocity from origin at start of observation is 47.5 m/s**2
distance from origin at start of observationis 10.0 m
Time after start of observation is 6.33 s
Distance from origin is 223.93 m