{
"cells": [
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# Chapter 19 Relative Motion"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# Example 19.1 Relative Velocity"
]
},
{
"cell_type": "code",
"execution_count": 2,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"The velocity at which the stone appears to hit the person travelling in the train is 11.180340 m/s\n",
"The direction of the stone is 26.565051 degree\n"
]
}
],
"source": [
"import math\n",
"# Initilization of variables\n",
"v_t=10 # m/s # velocity of the train\n",
"v_s=5 # m/s # velocity of the stone\n",
"# Calculations\n",
"# Let v_r be the relative velocity, which is given as, (from triangle law)\n",
"v_r=math.sqrt(v_t**2+v_s**2) # m/s\n",
"# The direction ofthe stone is,\n",
"theta=math.degrees(math.atan(v_s/v_t)) # degree\n",
"# Results\n",
"print('The velocity at which the stone appears to hit the person travelling in the train is %f m/s'%v_r)\n",
"print('The direction of the stone is %f degree'%theta)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# Example 19.2 Relative Velocity"
]
},
{
"cell_type": "code",
"execution_count": 12,
"metadata": {
"collapsed": false,
"scrolled": true
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"The magnitude of relative velocity of ship B with respect to ship A is 6.994832 m/s\n",
"The direction of the relative velocity is 14.638807 degree\n"
]
}
],
"source": [
"import math\n",
"# Initilization of variables\n",
"v_A=5 # m/s # speed of ship A\n",
"v_B=2.5 # m/s # speed of ship B\n",
"theta=135 # degree # angle between the two ships\n",
"# Calculations\n",
"# Here,\n",
"OA=v_A # m/s\n",
"OB=v_B # m/s\n",
"# The magnitude of relative velocity is given by cosine law as,\n",
"AB=math.sqrt((OA**2)+(OB**2)-(2*OA*OB*math.cos(theta*math.pi/180))) # m/s\n",
"# where AB gives the relative velocity of ship B with respect to ship A\n",
"# Applying sine law to find the direction, Let alpha be the direction of the reative velocity, then\n",
"alpha=math.degrees(math.asin((OB*math.sin(theta*math.pi/180))/(AB))) # degree\n",
"# Results\n",
"print('The magnitude of relative velocity of ship B with respect to ship A is %f m/s'%AB)\n",
"print('The direction of the relative velocity is %f degree'%alpha)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# Example 19.3 Relative Velocity"
]
},
{
"cell_type": "code",
"execution_count": 16,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"The magnitude of absolute velocity is 18.943769 km/hr\n",
"The direction of absolute velocity is 86.709116 degree\n"
]
}
],
"source": [
"import math\n",
"import numpy\n",
"# Initilization of variables\n",
"v_c=20 # km/hr # speed at which the cyclist is riding to west\n",
"theta_1=45 # degree # angle made by rain with the cyclist when he rides at 20 km/hr\n",
"V_c=12 # km/hr # changed speed\n",
"theta_2=30 # degree # changed angle when the cyclist rides at 12 km/hr\n",
"# Calculations\n",
"# Solving eq'ns 1 & 2 simultaneously to get the values of components(v_R_x & v_R_y) of absolute velocity v_R. We use matrix to solve eqn's 1 & 2.\n",
"A=numpy.matrix('1 1;1 0.577')\n",
"B=numpy.matrix('20;12')\n",
"C=numpy.linalg.inv(A)*B # km/hr\n",
"# The X component of relative velocity (v_R_x) is C(1)\n",
"# The Y component of relative velocity (v_R_y) is C(2)\n",
"# Calculations\n",
"# Relative velocity (v_R) is given as,\n",
"v_R=math.sqrt((C[0])**2+(C[1])**2) # km/hr\n",
"# And the direction of absolute velocity of rain is theta, is given as\n",
"theta=math.degrees(math.atan(C[1]/C[0])) # degree\n",
"# Results \n",
"print('The magnitude of absolute velocity is %f km/hr'%v_R)\n",
"print('The direction of absolute velocity is %f degree'%theta)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# Example 19.4 Relative Velocity"
]
},
{
"cell_type": "code",
"execution_count": 19,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"The relative position of car A relative to car B is 53.851648 m\n",
"The direction of car A w.r.t car B is 21.801409 degree\n",
"The velocity of car A relative to car B is 11.180340 m/s\n",
"The direction of car A w.r.t (for relative velocity)is 26.565051 degree\n",
"The acceleration of car A relative to car B is 1 m/s**2\n"
]
}
],
"source": [
"import math\n",
"# Initiization of variables\n",
"a=1 # m/s**2 # acceleration of car A\n",
"u_B=36*(1000/3600) # m/s # velocity of car B\n",
"u=0 # m/s # initial velocity of car A\n",
"d=32.5 # m # position of car A from north of crossing\n",
"t=5 # seconds\n",
"# Calculations\n",
"# CAR A: Absolute motion using eq'n v=u+at we have,\n",
"v=u+(a*t) # m/s\n",
"# Now distance travelled by car A after 5 seconds is given by, s_A=u*t+(1/2)*a*t**2\n",
"s_A=(u*t)+((1/2)*a*t**2)\n",
"# Now, let the position of car A after 5 seconds be y_A\n",
"y_A=d-s_A # m # \n",
"# CAR B:\n",
"# let a_B be the acceleration of car B\n",
"a_B=0 # m/s\n",
"# Now position of car B is s_B\n",
"s_B=(u_B*t)+((1/2)*a_B*t**2) # m\n",
"x_B=s_B # m\n",
"# Let the Relative position of car A with respect to car B be BA & its direction be theta, then from fig. 19.9(b)\n",
"OA=y_A\n",
"OB=x_B\n",
"BA=math.sqrt(OA**2+OB**2) # m\n",
"theta=math.degrees(math.atan(OA/OB)) # degree\n",
"# Let the relative velocity of car A w.r.t. the car B be v_AB & the angle be phi. Then from fig 19.9(c). Consider small alphabets\n",
"oa=v\n",
"ob=u_B\n",
"v_AB=math.sqrt(oa**2+ob**2) # m/s\n",
"phi=math.degrees(math.atan(oa/ob)) # degree\n",
"# Let the relative acceleration of car A w.r.t. car B be a_A/B.Then,\n",
"a_AB=a-a_B # m/s^2\n",
"# Results\n",
"print('The relative position of car A relative to car B is %f m'%BA)\n",
"print('The direction of car A w.r.t car B is %f degree'%theta)\n",
"print('The velocity of car A relative to car B is %f m/s'%v_AB)\n",
"print('The direction of car A w.r.t (for relative velocity)is %f degree'%phi)\n",
"print('The acceleration of car A relative to car B is %d m/s**2'%a_AB)"
]
}
],
"metadata": {
"anaconda-cloud": {},
"kernelspec": {
"display_name": "Python [Root]",
"language": "python",
"name": "Python [Root]"
},
"language_info": {
"codemirror_mode": {
"name": "ipython",
"version": 2
},
"file_extension": ".py",
"mimetype": "text/x-python",
"name": "python",
"nbconvert_exporter": "python",
"pygments_lexer": "ipython2",
"version": "2.7.12"
},
"widgets": {
"state": {},
"version": "1.1.2"
}
},
"nbformat": 4,
"nbformat_minor": 0
}