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diff --git a/Engineering_Mechanics/chapter_14.ipynb b/Engineering_Mechanics/chapter_14.ipynb new file mode 100644 index 00000000..6cceab74 --- /dev/null +++ b/Engineering_Mechanics/chapter_14.ipynb @@ -0,0 +1,1114 @@ +{
+ "metadata": {
+ "name": "chapter_14.ipynb"
+ },
+ "nbformat": 3,
+ "nbformat_minor": 0,
+ "worksheets": [
+ {
+ "cells": [
+ {
+ "cell_type": "heading",
+ "level": 1,
+ "metadata": {},
+ "source": [
+ "Chapter 14:Projectiles"
+ ]
+ },
+ {
+ "cell_type": "heading",
+ "level": 2,
+ "metadata": {},
+ "source": [
+ "Example 14.1,Page No.507 "
+ ]
+ },
+ {
+ "cell_type": "code",
+ "collapsed": false,
+ "input": [
+ "import math\n",
+ "from math import sin, cos, tan, radians, pi\n",
+ "\n",
+ "#Initilization of Variables\n",
+ "alpha=60 #Degrees\n",
+ "R=5000 #m #Horizontal Range\n",
+ "g=9.81 #m/s**2 #Acceleration due to gravity\n",
+ "\n",
+ "#Calculation\n",
+ "\n",
+ "#By equation of Horizontal Range,we get\n",
+ "u=((R*g)*(sin(2*alpha*pi*180**-1))**-1)**0.5 #m/s\n",
+ "\n",
+ "#Max Height attained by projectile\n",
+ "h_max=u**2*(sin(alpha*pi*180**-1))**2*(2*g)**-1\n",
+ "\n",
+ "#Result\n",
+ "print\"Velocity of projection is\",round(u,2),\"m/s\"\n",
+ "print\"Max Height attained by projectile is\",round(h_max,2),\"m\""
+ ],
+ "language": "python",
+ "metadata": {},
+ "outputs": [
+ {
+ "output_type": "stream",
+ "stream": "stdout",
+ "text": [
+ "Velocity of projection is 237.99 m/s\n",
+ "Max Height attained by projectile is 2165.06 m\n"
+ ]
+ }
+ ],
+ "prompt_number": 1
+ },
+ {
+ "cell_type": "heading",
+ "level": 2,
+ "metadata": {},
+ "source": [
+ "Example 14.2,Page No.507"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "collapsed": false,
+ "input": [
+ "import math\n",
+ "from math import sin, cos, tan, radians, pi\n",
+ "\n",
+ "#Initilization of Variables\n",
+ "u=100 #m/s #Velocity\n",
+ "alpha=30 #Degrees #Angle made by projectile with horizontal\n",
+ "g=9.81 #m/s #Acceleration due to gravity\n",
+ "\n",
+ "#Calculation\n",
+ "\n",
+ "#Horizontal Range\n",
+ "R=u**2*sin(2*alpha*pi*180**-1)*g**-1 #m\n",
+ "\n",
+ "#MAx height attained\n",
+ "h_max=u**2*sin(alpha*pi*180**-1)*g**-1 #m\n",
+ "\n",
+ "#Time of flight\n",
+ "T=2*u*sin(alpha*pi*180**-1)*g**-1 #s\n",
+ "\n",
+ "#Result\n",
+ "print\"Horizontal Range is\",round(R,2),\"m\"\n",
+ "print\"MAx Height attained is\",round(h_max,2),\"m\"\n",
+ "print\"Time of Flight is\",round(T,2),\"s\" "
+ ],
+ "language": "python",
+ "metadata": {},
+ "outputs": [
+ {
+ "output_type": "stream",
+ "stream": "stdout",
+ "text": [
+ "Horizontal Range is 882.8 m\n",
+ "MAx Height attained is 509.68 m\n",
+ "Time of Flight is 10.19 s\n"
+ ]
+ }
+ ],
+ "prompt_number": 2
+ },
+ {
+ "cell_type": "heading",
+ "level": 2,
+ "metadata": {},
+ "source": [
+ "Example 14.3,Page No.508"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "collapsed": false,
+ "input": [
+ "import math\n",
+ "from math import sin, cos, tan, radians, pi\n",
+ "import numpy as np\n",
+ "\n",
+ "#Initilization of Variables\n",
+ "#R=4*h_max\n",
+ "\n",
+ "#Calculation\n",
+ "\n",
+ "#Equation ofhorizontal Range is\n",
+ "#R=u**2*sin(2*alpha)*g**-1 .........1\n",
+ "\n",
+ "#Equation of MAx height \n",
+ "#h_max=u**2*sin(alpha)**2*(2*g)**-1 .........2\n",
+ "\n",
+ "#After simplifying both equations,we get\n",
+ "alpha=np.arctan(1)*(pi**-1*180)\n",
+ "\n",
+ "#Result\n",
+ "print\"Angle of projections is\",round(alpha,2),\"degrees\""
+ ],
+ "language": "python",
+ "metadata": {},
+ "outputs": [
+ {
+ "output_type": "stream",
+ "stream": "stdout",
+ "text": [
+ "Angle of projections is 45.0 degrees\n"
+ ]
+ }
+ ],
+ "prompt_number": 3
+ },
+ {
+ "cell_type": "heading",
+ "level": 2,
+ "metadata": {},
+ "source": [
+ "Example 14.4,Page No.508"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "collapsed": false,
+ "input": [
+ "import math\n",
+ "from math import sin, cos, tan, radians, pi\n",
+ "import numpy as np\n",
+ "\n",
+ "#Initilization of Variables\n",
+ "u=20 #m/s #Velocity\n",
+ "x=20 #m #X cordinate of trajectory\n",
+ "y=8 #m #Y cordinate of trajectory\n",
+ "\n",
+ "#Calculation\n",
+ "\n",
+ "#equation of trajectory\n",
+ "#y=x*tan(alpha)-g*x**2*(2*u**2*cos(alpha)**2)**-1\n",
+ "#After substituting values and further simplifying we get quadratic equation\n",
+ "#4.905*(tan(alpha))**2-20*tan(alpha)+12.905=0\n",
+ "a=4.905\n",
+ "b=-20\n",
+ "c=12.905\n",
+ "\n",
+ "X=b**2-4*a*c\n",
+ "\n",
+ "y1=(-b+X**0.5)*(2*a)**-1\n",
+ "y2=(-b-X**0.5)*(2*a)**-1\n",
+ "\n",
+ "alpha1=np.arctan(y1)*(pi**-1*180) #Degrees\n",
+ "alpha2=np.arctan(y2)*(pi**-1*180) #Degrees\n",
+ "\n",
+ "#Result\n",
+ "print\"Angle of projection of particle is:alpha1\",round(alpha1,2),\"Degrees\"\n",
+ "print\" :alpha2\",round(alpha2,2),\"Degrees\""
+ ],
+ "language": "python",
+ "metadata": {},
+ "outputs": [
+ {
+ "output_type": "stream",
+ "stream": "stdout",
+ "text": [
+ "Angle of projection of particle is:alpha1 73.01 Degrees\n",
+ " :alpha2 38.79 Degrees\n"
+ ]
+ }
+ ],
+ "prompt_number": 2
+ },
+ {
+ "cell_type": "heading",
+ "level": 2,
+ "metadata": {},
+ "source": [
+ "Example 14.5,Page No.509"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "collapsed": false,
+ "input": [
+ "import math\n",
+ "from math import sin, cos, tan, radians, pi\n",
+ "\n",
+ "#Initilization of Variables\n",
+ "\n",
+ "x=4.8 #m #X-cord of projectile\n",
+ "y=3.6 #m #y-cord of projectile\n",
+ "alpha=60 #Degrees #Inclination of jet\n",
+ "g=9.81 #m/s**2\n",
+ "\n",
+ "#Calculation\n",
+ "\n",
+ "#Equation of trajectory\n",
+ "#y=x*tan(alpha)-g*x**2*(2*u**2*cos(alpha)**2)**-1\n",
+ "#After further simplifying we get\n",
+ "u=((g*x**2)*(2*(cos(alpha*pi*180**-1))**2*((x*tan(alpha*pi*180**-1))-y))**-1)**0.5 #m/s\n",
+ "\n",
+ "#Result\n",
+ "print\"required velocity of jet at nozzle exit is\",round(u,2),\"m/s\""
+ ],
+ "language": "python",
+ "metadata": {},
+ "outputs": [
+ {
+ "output_type": "stream",
+ "stream": "stdout",
+ "text": [
+ "required velocity of jet at nozzle exit is 9.79 m/s\n"
+ ]
+ }
+ ],
+ "prompt_number": 5
+ },
+ {
+ "cell_type": "heading",
+ "level": 2,
+ "metadata": {},
+ "source": [
+ "Example 14.6,Page No.510"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "collapsed": false,
+ "input": [
+ "import math\n",
+ "from math import sin, cos, tan, radians, pi\n",
+ "import numpy as np\n",
+ "\n",
+ "#Initilization of Variables\n",
+ "\n",
+ "u=250 #m/s #Velocity\n",
+ "x=4000 #m #x-cord\n",
+ "y=700 #m #y-cord\n",
+ "\n",
+ "#Calculation\n",
+ "\n",
+ "#equation of trajectory\n",
+ "#y=x*tan(alpha)-g*x**2*(2*u**2*cos(alpha)**2)**-1\n",
+ "#After substituting values and further simplifying we get quadratic equation\n",
+ "#1255.68*tan(alpha)**2-4000*tan(alpha)+1955.68=0\n",
+ "a=1255.68\n",
+ "b=-4000\n",
+ "c=1955.68\n",
+ "\n",
+ "X=b**2-4*a*c\n",
+ "\n",
+ "y1=(-b+X**0.5)*(2*a)**-1\n",
+ "y2=(-b-X**0.5)*(2*a)**-1\n",
+ "\n",
+ "alpha1=np.arctan(y1)*(pi**-1*180) #Degrees\n",
+ "alpha2=np.arctan(y2)*(pi**-1*180) #Degrees\n",
+ "\n",
+ "#Result\n",
+ "print\"Firing angle to hit the target is:alpha1\",round(alpha1,2),\"Degrees\"\n",
+ "print\" :alpha2\",round(alpha2,2),\"Degrees\""
+ ],
+ "language": "python",
+ "metadata": {},
+ "outputs": [
+ {
+ "output_type": "stream",
+ "stream": "stdout",
+ "text": [
+ "Firing angle to hit the target is:alpha1 68.83 Degrees\n",
+ " :alpha2 31.09 Degrees\n"
+ ]
+ }
+ ],
+ "prompt_number": 6
+ },
+ {
+ "cell_type": "heading",
+ "level": 2,
+ "metadata": {},
+ "source": [
+ "Example 14.7,Page No.511"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "collapsed": false,
+ "input": [
+ "import math\n",
+ "from math import sin, cos, tan, radians, pi\n",
+ "import numpy as np\n",
+ "\n",
+ "#Initilization of Variables\n",
+ "\n",
+ "alpha1=15 #Degrees #Angle of projectile 1\n",
+ "alpha2=45 #Degrees #Angle of projectile2\n",
+ "g=9.81 #m/s**2\n",
+ "#R1=R-12\n",
+ "#R2=R+24\n",
+ "\n",
+ "#Calculation\n",
+ "\n",
+ "#form Equation of horizontal Range we get,\n",
+ "#R-12=u**2*sin(2*alpha1)*g**-1 ..................1\n",
+ "\n",
+ "#R+24=u**2*sin(2*alpha2)*g**-1 ....................2\n",
+ "\n",
+ "#Dividing equation 1 by 2 and further simplifying we get\n",
+ "R=24+24 #m\n",
+ "\n",
+ "#Sub value of R in equation 2 we get\n",
+ "#u**2=g*72\n",
+ "\n",
+ "#Sub values of R and u**2 in equation of horizontal range we get\n",
+ "#sin(2*alpha)=R*g*(g*72)**-1\n",
+ "#LEt sin(2*alpha)=X\n",
+ "X=R*g*(g*72)**-1\n",
+ "alpha=(np.arcsin(X)*(180*pi**-1))*2**-1\n",
+ "\n",
+ "#Result\n",
+ "print\"Angle of projection to hit the mark is\",round(alpha,2)"
+ ],
+ "language": "python",
+ "metadata": {},
+ "outputs": [
+ {
+ "output_type": "stream",
+ "stream": "stdout",
+ "text": [
+ "Angle of projection to hit the mark is 20.91\n"
+ ]
+ }
+ ],
+ "prompt_number": 7
+ },
+ {
+ "cell_type": "heading",
+ "level": 2,
+ "metadata": {},
+ "source": [
+ "Example 14.8,Page No.512"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "collapsed": false,
+ "input": [
+ "import math\n",
+ "from math import sin, cos, tan, radians, pi\n",
+ "import numpy as np\n",
+ "\n",
+ "#Initilization of Variables\n",
+ "\n",
+ "u=100 #m/s #initial Velocity\n",
+ "alpha=30 #DEgrees #Angle of projection\n",
+ "g=9.81 #m/s**2\n",
+ "h=80 #m #Height below B\n",
+ "\n",
+ "#Calculation\n",
+ "\n",
+ "#Max Height \n",
+ "H=u**2*(sin(alpha*pi*180**-1))**2*(2*g)**-1 #m\n",
+ "\n",
+ "#Vertical Distance\n",
+ "S=H+h #m\n",
+ "\n",
+ "#Vertical Component of velocity striking the target\n",
+ "v2=(2*g*S)**0.5 #m/s\n",
+ "\n",
+ "#Horizontal component of velocity\n",
+ "v=u*cos(alpha*pi*180**-1)\n",
+ "\n",
+ "#Actual Velocity with which bullet strikes the target\n",
+ "V=(v2**2+v**2)**0.5\n",
+ "\n",
+ "#Angle made by actual velocity striking velocity\n",
+ "theta=np.arctan(v2*v**-1)*(pi**-1*180)\n",
+ "\n",
+ "#Result\n",
+ "print\"Max height attained by bullet\",round(H,2),\"m\"\n",
+ "print\"Actual Velocity with which it will strike the target\",round(theta,2),\"Degrees\""
+ ],
+ "language": "python",
+ "metadata": {},
+ "outputs": [
+ {
+ "output_type": "stream",
+ "stream": "stdout",
+ "text": [
+ "Max height attained by bullet 127.42 m\n",
+ "Actual Velocity with which it will strike the target 36.38 Degrees\n"
+ ]
+ }
+ ],
+ "prompt_number": 8
+ },
+ {
+ "cell_type": "heading",
+ "level": 2,
+ "metadata": {},
+ "source": [
+ "Example 14.9,Page No.514"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "collapsed": false,
+ "input": [
+ "import math\n",
+ "from math import sin, cos, tan, radians, pi\n",
+ "import numpy as np\n",
+ "\n",
+ "#Initilization of Variables\n",
+ "h=150 #m #Height of cliff\n",
+ "u=180 #m/s #Initial Velocity\n",
+ "alpha=30 #ANgle of projection\n",
+ "g=9.81\n",
+ "\n",
+ "#Calculation\n",
+ "\n",
+ "#From equation of vertical Distance\n",
+ "#y=u*sin(Alpha)*t-0.5*g*t**2\n",
+ "#Sub values and further simplifying we get\n",
+ "#4.905t**2-90*t-150=0\n",
+ "a=4.905 \n",
+ "b=-90\n",
+ "c=-150\n",
+ "\n",
+ "X=(b**2-4*a*c)**0.5\n",
+ "\n",
+ "#Total time of flight\n",
+ "t=(-b+(X))*(2*a)**-1\n",
+ "\n",
+ "#Horizontal Distance \n",
+ "h1=u*cos(alpha*pi*180**-1)*t\n",
+ "\n",
+ "#MAx Height \n",
+ "H=u**2*sin(alpha*pi*180**-1)**2*(2*g)**-1 #m\n",
+ "\n",
+ "#ELEvation above the ground\n",
+ "H2=h+H #m\n",
+ "\n",
+ "#Result\n",
+ "print\"Horizontal Distance from gun is\",round(h1,2),\"m\"\n",
+ "print\"Elevation above the ground\",round(H2,2),\"m\""
+ ],
+ "language": "python",
+ "metadata": {},
+ "outputs": [
+ {
+ "output_type": "stream",
+ "stream": "stdout",
+ "text": [
+ "Horizontal Distance from gun is 3099.98 m\n",
+ "Elevation above the ground 562.84 m\n"
+ ]
+ }
+ ],
+ "prompt_number": 3
+ },
+ {
+ "cell_type": "heading",
+ "level": 2,
+ "metadata": {},
+ "source": [
+ "Example 14.10,Page No.515"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "collapsed": false,
+ "input": [
+ "import math\n",
+ "from math import sin, cos, tan, radians, pi\n",
+ "import numpy as np\n",
+ "\n",
+ "#Initilization of Variables\n",
+ "\n",
+ "h=6 #m #Height of tunnel\n",
+ "u=50 #m/s #Initial Velocity\n",
+ "g=9.81\n",
+ "\n",
+ "#Calculation\n",
+ "\n",
+ "alpha=np.arcsin(((h*2*g)*((u**2)**-1))**0.5)*(pi**-1*180)\n",
+ "\n",
+ "#Horizontal Range \n",
+ "R=u**2*sin(2*alpha*pi*180**-1)*g**-1\n",
+ "\n",
+ "#Result\n",
+ "print\"Angle of projection is\",round(alpha,2),\"DEgrees\"\n",
+ "print\"Horizontal Range is\",round(R,2),\"m\""
+ ],
+ "language": "python",
+ "metadata": {},
+ "outputs": [
+ {
+ "output_type": "stream",
+ "stream": "stdout",
+ "text": [
+ "Angle of projection is 12.53 DEgrees\n",
+ "Horizontal Range is 107.96 m\n"
+ ]
+ }
+ ],
+ "prompt_number": 10
+ },
+ {
+ "cell_type": "heading",
+ "level": 2,
+ "metadata": {},
+ "source": [
+ "Example 14.11,Page No.516"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "collapsed": false,
+ "input": [
+ "import math\n",
+ "from math import sin, cos, tan, radians, pi\n",
+ "\n",
+ "#Initilization of Variables\n",
+ "alpha1=30 #Degrees #Angle of projection1\n",
+ "alpha2=30 #Degrees #Angle of projection2\n",
+ "u1=350 #m/s #Velocity of projection at A\n",
+ "u2=300 #m/s #Velocity of projection at B\n",
+ "L_AB=30 #m #distance between A nd B\n",
+ "g=9.81 #m/s**2 \n",
+ "\n",
+ "#Calculation\n",
+ "\n",
+ "#Length AD\n",
+ "L_AD=L_AB*cos(alpha1*pi*180**-1)\n",
+ "\n",
+ "#Horizontal component of velocity at A\n",
+ "V_A=u1*cos(alpha1*pi*180) #m/s\n",
+ "\n",
+ "#Horizontal component of velocity at B\n",
+ "V_B=u2*cos(alpha2*pi*180) #m/s\n",
+ "\n",
+ "#Horizontal Distance covered by shot A\n",
+ "#x=(u1*cos(alpha1)*t ..........................1\n",
+ "\n",
+ "#Horizontal Distance covered by shot B\n",
+ "#15*(3)**0.5-x=(u2*cos(alpha2))*t ................2\n",
+ "\n",
+ "#Adding Equations 1 and 2 we get\n",
+ "t=15*(3)**0.5*((u1+u2)*cos(alpha1*pi*180**-1))**-1\n",
+ "\n",
+ "#sub values in equation 1 we get\n",
+ "x=(u1*cos(alpha1*pi*180**-1))*t\n",
+ "\n",
+ "#Intial velocity shot in vertical direction\n",
+ "V2=u1*sin(alpha1*pi*180**-1) #m/s\n",
+ "\n",
+ "#Distance in vertical direction\n",
+ "y=V2*t-g*t**2*2**-1\n",
+ "\n",
+ "#Result\n",
+ "print\"Time when these two shots meet\",round(t,2),\"s\"\n",
+ "print\"Vertical Distance at which they meet\",round(y,3),\"m\""
+ ],
+ "language": "python",
+ "metadata": {},
+ "outputs": [
+ {
+ "output_type": "stream",
+ "stream": "stdout",
+ "text": [
+ "Time when these two shots meet 0.05 s\n",
+ "Vertical Distance at which they meet 8.066 m\n"
+ ]
+ }
+ ],
+ "prompt_number": 11
+ },
+ {
+ "cell_type": "heading",
+ "level": 2,
+ "metadata": {},
+ "source": [
+ "Example 14.12,Page No.518"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "collapsed": false,
+ "input": [
+ "import math\n",
+ "\n",
+ "#Initilization of Variables\n",
+ "v=720*10**3*3600**-1 #m/s #Speed of air craft\n",
+ "u=200 #m/s #Velocity of bomb\n",
+ "H=1000 #m #Height\n",
+ "g=9.81\n",
+ "\n",
+ "#Calculation\n",
+ "\n",
+ "#Time requaired by bomb\n",
+ "t=(H*2*g**-1)**0.5 #s\n",
+ "\n",
+ "#Horizontal distance of air craft\n",
+ "h=u*round(t,3) #m\n",
+ "\n",
+ "#Result\n",
+ "print\"Time requaired by bomb to reach the ground\",round(t,2),\"s\"\n",
+ "print\"Horizontal distance of air craft\",round(h,2),\"m\""
+ ],
+ "language": "python",
+ "metadata": {},
+ "outputs": [
+ {
+ "output_type": "stream",
+ "stream": "stdout",
+ "text": [
+ "Time requaired by bomb to reach the ground 14.28 s\n",
+ "Horizontal distance of air craft 2855.6 m\n"
+ ]
+ }
+ ],
+ "prompt_number": 4
+ },
+ {
+ "cell_type": "heading",
+ "level": 2,
+ "metadata": {},
+ "source": [
+ "Example 14.13,Page No.519"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "collapsed": false,
+ "input": [
+ "import math\n",
+ "from math import sin, cos, tan, radians, pi\n",
+ "import numpy as np\n",
+ "\n",
+ "#Initilization of Variables\n",
+ "\n",
+ "v=108*10**3*3600**-1 #m/s #Spedd of air craft\n",
+ "u=30 #m/s #Horizontal Velocity\n",
+ "H=1000 #m #Height\n",
+ "g=9.81 #m/s**2\n",
+ "\n",
+ "#Calculation\n",
+ "\n",
+ "#Time taken by bomb\n",
+ "t=(H*2*g**-1)**0.5 #s\n",
+ "\n",
+ "#Horizontal Distance OB\n",
+ "h=u*t #m\n",
+ "\n",
+ "#Velocity of bomb hitting the ground\n",
+ "V=g*t #m/s**2\n",
+ "\n",
+ "#Resultant Velocity\n",
+ "V2=(u**2+V**2)**0.5 #m/s\n",
+ "\n",
+ "#Direction in which the bomb hits the ground\n",
+ "theta=np.arctan(V*u**-1)*(180*pi**-1)\n",
+ "\n",
+ "#Result\n",
+ "print\"HOrizontal Distance of air craft from target\",round(h,2),\"m\"\n",
+ "print\"Velocity of bomb\",round(V2,2),\"m/s\"\n",
+ "print\"Direction of bomb\",round(theta,2),\"Degrees\""
+ ],
+ "language": "python",
+ "metadata": {},
+ "outputs": [
+ {
+ "output_type": "stream",
+ "stream": "stdout",
+ "text": [
+ "HOrizontal Distance of air craft from target 428.35 m\n",
+ "Velocity of bomb 143.25 m/s\n",
+ "Direction of bomb 77.91 Degrees\n"
+ ]
+ }
+ ],
+ "prompt_number": 13
+ },
+ {
+ "cell_type": "heading",
+ "level": 2,
+ "metadata": {},
+ "source": [
+ "Example 14.14,Page No.520"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "collapsed": false,
+ "input": [
+ "import math\n",
+ "\n",
+ "#Initilization of Variables\n",
+ "H=2000 #m #Height of aeroplane\n",
+ "u=10**6*3600**-1 #Velocity of plane #m/s\n",
+ "g=9.8 #m/s**2\n",
+ "\n",
+ "#Calculation\n",
+ "\n",
+ "#Time required by bomb\n",
+ "t=(2*H*g**-1)**0.5 #s\n",
+ "\n",
+ "#Horizontal Distance travelled by bomb\n",
+ "h=u*t #m\n",
+ "\n",
+ "#Result\n",
+ "print\"Time required by bomb to reach the ground\",round(t,2),\"s\"\n",
+ "print\"Horizontal Distance travelled by bomb\",round(h,2),\"m\""
+ ],
+ "language": "python",
+ "metadata": {},
+ "outputs": [
+ {
+ "output_type": "stream",
+ "stream": "stdout",
+ "text": [
+ "Time required by bomb to reach the ground 20.2 s\n",
+ "Horizontal Distance travelled by bomb 5611.96 m\n"
+ ]
+ }
+ ],
+ "prompt_number": 14
+ },
+ {
+ "cell_type": "heading",
+ "level": 2,
+ "metadata": {},
+ "source": [
+ "Example 14.16,Page No.521"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "collapsed": false,
+ "input": [
+ "import math\n",
+ "from math import sin, cos, tan, radians, pi\n",
+ "import numpy as np\n",
+ "\n",
+ "#Initilization of Variables\n",
+ "L_AB1=4 #m #LEngth of AB in Horizontal Distance\n",
+ "L_AB2=2 #m #LEngth of AB in Vertical Distance\n",
+ "g=9.81 #m/s**2 \n",
+ "\n",
+ "#Calculation\n",
+ "\n",
+ "#time\n",
+ "t=((L_AB2*2)*(9.81)**-1)**0.5\n",
+ "\n",
+ "#Minimum veloacity of motorcyclye at A in Horizontal Direction\n",
+ "u=L_AB1*t**-1 #m/s\n",
+ "\n",
+ "#Vertical Component of velocity\n",
+ "v=g*t #m/s\n",
+ "\n",
+ "#Resultant velocity at B\n",
+ "V=(u**2+v**2)**0.5*3600*(10**3)**-1 #km/hr\n",
+ "\n",
+ "#Inclination\n",
+ "theta=np.arctan(v*u**-1)*(180*pi**-1)\n",
+ "\n",
+ "#Result\n",
+ "print\"Inclination of motorcycle clearing the ditch\",round(theta,2),\"Degrees\"\n",
+ "print\"MAgnitude of velocity of motorcycleafter clearing ditch\",round(V,2),\"km/hr\""
+ ],
+ "language": "python",
+ "metadata": {},
+ "outputs": [
+ {
+ "output_type": "stream",
+ "stream": "stdout",
+ "text": [
+ "Inclination of motorcycle clearing the ditch 45.0 Degrees\n",
+ "MAgnitude of velocity of motorcycleafter clearing ditch 31.89 km/hr\n"
+ ]
+ }
+ ],
+ "prompt_number": 15
+ },
+ {
+ "cell_type": "heading",
+ "level": 2,
+ "metadata": {},
+ "source": [
+ "Example 14.17,Page No.522"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "collapsed": false,
+ "input": [
+ "import math\n",
+ "from math import sin, cos, tan, radians, pi\n",
+ "\n",
+ "#Initilization of Variables\n",
+ "alpha=25 #Degrees #Angle of ramp\n",
+ "L_BC1=4 #m #Horizontal Distance between B and C\n",
+ "L_BC2=2 #m #vertical Distance between B and C\n",
+ "g=9.81 #m/s**2\n",
+ "\n",
+ "#Calculation\n",
+ "\n",
+ "##From equation of path \n",
+ "X=(L_BC2+L_BC1*tan(alpha*pi*180**-1))\n",
+ "Y=(g*L_BC1**2)*(2*(cos(alpha*pi*180**-1))**2)**-1\n",
+ "u=(X**-1*Y)**0.5\n",
+ "\n",
+ "#Result\n",
+ "print\"Minimum speed of motorcycle is\",round(u,2),\"m/s\""
+ ],
+ "language": "python",
+ "metadata": {},
+ "outputs": [
+ {
+ "output_type": "stream",
+ "stream": "stdout",
+ "text": [
+ "Minimum speed of motorcycle is 4.97 m/s\n"
+ ]
+ }
+ ],
+ "prompt_number": 5
+ },
+ {
+ "cell_type": "heading",
+ "level": 2,
+ "metadata": {},
+ "source": [
+ "Example 14.18,Page No.526"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "collapsed": false,
+ "input": [
+ "import math\n",
+ "from math import sin, cos, tan, radians, pi\n",
+ "\n",
+ "#Initilization of Variables\n",
+ "u=40 #m/s #Velocity of projection\n",
+ "alpha=50 #Degrees #Angle of projection\n",
+ "beta=20 #degrees #Ang;e if incline plane\n",
+ "g=9.81\n",
+ "\n",
+ "#Calculation\n",
+ "\n",
+ "#Time of flight\n",
+ "T=2*u*sin((alpha-beta)*pi*180**-1)*(g*cos(beta*pi*180**-1))**-1\n",
+ "\n",
+ "#Range \n",
+ "R=2*u**2*cos(alpha*pi*180**-1)*sin((alpha-beta)*pi*180**-1)*(g*(cos(beta*pi*180**-1))**2)**-1\n",
+ "\n",
+ "#Let alpha2 be the angle of projection for max range\n",
+ "alpha2=(90+beta)*2**-1\n",
+ "\n",
+ "#MAx range up the plane\n",
+ "R2=u**2*(g*(1+sin(beta*pi*180**-1)))**-1\n",
+ "\n",
+ "#Result\n",
+ "print\"Time of flight is\",round(T,2),\"s\"\n",
+ "print\"range up the plane is\",round(R,2),\"m\"\n",
+ "print\"Max range up the plane\",round(R2,2),\"m\""
+ ],
+ "language": "python",
+ "metadata": {},
+ "outputs": [
+ {
+ "output_type": "stream",
+ "stream": "stdout",
+ "text": [
+ "Time of flight is 4.34 s\n",
+ "range up the plane is 118.73 m\n",
+ "Max range up the plane 121.53 m\n"
+ ]
+ }
+ ],
+ "prompt_number": 6
+ },
+ {
+ "cell_type": "heading",
+ "level": 2,
+ "metadata": {},
+ "source": [
+ "Example 14.19,Page No.527"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "collapsed": false,
+ "input": [
+ "import math\n",
+ "from math import sin, cos, tan, radians, pi\n",
+ "\n",
+ "#Initilization of Variables\n",
+ "\n",
+ "u=30 #m/s #velocity\n",
+ "alpha=55 #angle of projection\n",
+ "beta=20 #angle of plane\n",
+ "g=9.81\n",
+ "\n",
+ "#Calculation\n",
+ "#Maximum Range\n",
+ "R=u**2*(g*(1+sin(beta*pi*180**-1)))**-1 #m\n",
+ "\n",
+ "#Time of Flight\n",
+ "T=2*u*sin((alpha-beta)*pi*180**-1)*(g*cos(beta*pi*180**-1))**-1\n",
+ "\n",
+ "#Result\n",
+ "print\"Time of Fliight is\",round(T,2),\"s\"\n",
+ "print\"Maximum Range is \",round(R,2),\"m\""
+ ],
+ "language": "python",
+ "metadata": {},
+ "outputs": [
+ {
+ "output_type": "stream",
+ "stream": "stdout",
+ "text": [
+ "Time of Fliight is 3.73 s\n",
+ "Maximum Range is 68.36 m\n"
+ ]
+ }
+ ],
+ "prompt_number": 7
+ },
+ {
+ "cell_type": "heading",
+ "level": 2,
+ "metadata": {},
+ "source": [
+ "Example 14.20,Page No.527"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "collapsed": false,
+ "input": [
+ "import math\n",
+ "from math import sin, cos, tan, radians, pi\n",
+ "\n",
+ "#Initilization of Variables\n",
+ "\n",
+ "u=3 #m/s #Velocity of projection\n",
+ "alpha=25 #Degrees #Angle of projection\n",
+ "beta=25 #degrees #Angle of plane with horizontal\n",
+ "g=9.81 #m/s**2\n",
+ "\n",
+ "#Calculation\n",
+ "\n",
+ "#Range\n",
+ "R=2*u**2*cos(alpha*pi*180**-1)*sin((alpha+beta)*pi*180**-1)*(g*(cos(beta*pi*180**-1))**2)**-1\n",
+ "\n",
+ "#L_BC=y\n",
+ "#L_AC=x\n",
+ "\n",
+ "#From tan(beta) we get\n",
+ "#y=0.466*x .................(1)\n",
+ "\n",
+ "#From Equation (L_AB)**2=(L_BC**2+L_AC)**2\n",
+ "#After sub values and further simplifying we get\n",
+ "x=(2.4*(1.217)**-1)**0.5 #m\n",
+ "\n",
+ "#Sub in equation 1 we get\n",
+ "y=0.466*x\n",
+ "\n",
+ "#Result\n",
+ "print\"Co-ordinates of point B are:x\",round(x,2),\"m\"\n",
+ "print\" :y\",round(y,2),\"m\""
+ ],
+ "language": "python",
+ "metadata": {},
+ "outputs": [
+ {
+ "output_type": "stream",
+ "stream": "stdout",
+ "text": [
+ "Co-ordinates of point B are:x 1.4 m\n",
+ " :y 0.65 m\n"
+ ]
+ }
+ ],
+ "prompt_number": 12
+ },
+ {
+ "cell_type": "heading",
+ "level": 2,
+ "metadata": {},
+ "source": [
+ "Example 14.21,Page No.528"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "collapsed": false,
+ "input": [
+ "import math\n",
+ "from math import sin, cos, tan, radians, pi\n",
+ "import numpy as np\n",
+ "\n",
+ "#Initilization of Variables\n",
+ "\n",
+ "L_AB=75 #m #LEngth of AB\n",
+ "h=19.6 #m #Height of point of release\n",
+ "beta=26.564 #Degrees #angle of plane \n",
+ "\n",
+ "#Calculation\n",
+ "\n",
+ "#Length AC\n",
+ "L_AC=L_AB*cos(beta*pi*180**-1) #m\n",
+ "\n",
+ "#From horizontal Distance \n",
+ "#u*cos(alpha)=L_AC*t**-1 .............1\n",
+ "\n",
+ "#Vertical motion motion from point of release\n",
+ "#u*sin(alpha)=19.6 ....................2\n",
+ "\n",
+ "#vertical Distance\n",
+ "y=L_AB*sin(beta*pi*180**-1) #m\n",
+ "\n",
+ "#Vertical distance from point of release travelled by ball\n",
+ "#y=u*sin(alpha)*t-0.5*g*t**2\n",
+ "\n",
+ "#sub value of y in above equation and further simplifyin we get\n",
+ "#t**2-4t-6.84=0\n",
+ "a=1\n",
+ "b=-4\n",
+ "c=-6.84\n",
+ "\n",
+ "X=b**2-4*a*c\n",
+ "\n",
+ "t=(-b+X**0.5)*(2*a)**-1\n",
+ "\n",
+ "#sub value in equation 1 we get\n",
+ "#Let ucos(alpha)=X\n",
+ "X=67.08*t**-1 #..........................3\n",
+ "\n",
+ "#Dividing equation 1 by 3 we get\n",
+ "alpha=np.arctan(19.6*12.68**-1)*(180*pi**-1)\n",
+ "\n",
+ "#sub in equation 1 we get\n",
+ "u=12.68*cos(alpha*pi*180**-1)**-1\n",
+ "\n",
+ "#Result\n",
+ "print\"Initial Velocity\",round(u,2),\"m/s\"\n",
+ "print\"Inclination is\",round(alpha,4),\"Degrees\""
+ ],
+ "language": "python",
+ "metadata": {},
+ "outputs": [
+ {
+ "output_type": "stream",
+ "stream": "stdout",
+ "text": [
+ "Initial Velocity 23.34 m/s\n",
+ "Inclination is 57.0996 Degrees\n"
+ ]
+ }
+ ],
+ "prompt_number": 9
+ }
+ ],
+ "metadata": {}
+ }
+ ]
+}
\ No newline at end of file |