Position, Velocity, Acceleration Vectors Calc IV Lab Karen Donnelly All Rights Reserved LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEocmVzdGFydEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIjpGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1GNjYtUSJ+RidGOUY7Rj5GQEZCRkRGRkZIL0ZLUSYwLjBlbUYnL0ZORlM= QyYtSSV3aXRoRzYiNiNJJnBsb3RzRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlISIiLUYkNiNJKnBsb3R0b29sc0dGKEYr QyQtSSV3aXRoRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiNJL1ZlY3RvckNhbGN1bHVzR0YlISIi The following is to convince Maple that the variable t is only real-valued (not complex). QyYtSSdhc3N1bWVHNiI2IydJInRHRiVJJXJlYWxHRiUhIiItSSppbnRlcmZhY2VHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiU2Iy9JLHNob3dhc3N1bWVkR0YlIiIhRio=

Consider vector-valued function r(t) representing position in space as a function of time t. The velocity vector v(t) (also called the tangent vector) at point P = r(LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiNRIUYnLyUrYmFja2dyb3VuZEdRLlsyNTUsMjU1LDI1NV1GJy9GNlEnbm9ybWFsRicvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0Y9RkA=) is given by v(LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiNRIUYnLyUrYmFja2dyb3VuZEdRLlsyNTUsMjU1LDI1NV1GJy9GNlEnbm9ybWFsRicvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0Y9RkA=) = r'(LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiNRIUYnLyUrYmFja2dyb3VuZEdRLlsyNTUsMjU1LDI1NV1GJy9GNlEnbm9ybWFsRicvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0Y9RkA=). The direction of this vector is the direction of motion at time LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiMEYnL0Y2USdub3JtYWxGJy8lK2JhY2tncm91bmRHUS5bMjU1LDI1NSwyNTVdRidGPi8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnRkBGPg==, and its magnitude || r'(LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiMEYnL0Y2USdub3JtYWxGJy8lK2JhY2tncm91bmRHUS5bMjU1LDI1NSwyNTVdRidGPi8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnRkBGPg==)|| is the speed of the object at time LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiMEYnL0Y2USdub3JtYWxGJy8lK2JhY2tncm91bmRHUS5bMjU1LDI1NSwyNTVdRidGPi8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnRkBGPg==. It also gives use the direction vector for the tangent line to the curve represented by r(t). The acceleration vector a for the position vector is given by a(t) = r''(t) = v'(t). It gives a measure of the rate of change of velocity -- i.e. rate at which the direction and speed of motion are changing. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

As an example consider a helix curve. We create R using the PositionVector function from the VectorCalculus package. PkkiUkc2Ii1JL1Bvc2l0aW9uVmVjdG9yR0YkNiM3JS1JIipHNiQlKnByb3RlY3RlZEcvSSttb2R1bGVuYW1lR0YkSS9WZWN0b3JDYWxjdWx1c0c2JEYsSShfc3lzbGliR0YkNiQiIiUtSSRzaW5HRiQ2I0kidEdGJC1GKjYkRjMtSSRjb3NHRiRGNkY3 QyU+SSZQbG90Ukc2Ii1JM1Bsb3RQb3NpdGlvblZlY3RvckdGJTYlSSJSR0YlL0kidEdGJTsiIiEtSSIqRzYkJSpwcm90ZWN0ZWRHL0krbW9kdWxlbmFtZUdGJUkvVmVjdG9yQ2FsY3VsdXNHNiRGMUkoX3N5c2xpYkdGJTYkIiIlSSNQaUdGMS9JLWN1cnZlb3B0aW9uc0dGJTclL0kmY29sb3JHRiVJJHJlZEdGJS9JJWF4ZXNHRiVJJmJveGVkR0YlL0knbGFiZWxzR0YlNyVJInhHRiVJInlHRiVJInpHRiUhIiJGJA==

The velocity vector function is found by differentiating the position vector with respect to t. PkkiVkc2Ii1JJWRpZmZHJSpwcm90ZWN0ZWRHNiRJIlJHRiRJInRHRiQ= If we evaluate the position vector at time t = 2.0, we will have the position at that time: (using Maple's subs command) PkkjUjFHNiItSSVzdWJzRyUqcHJvdGVjdGVkRzYkL0kidEdGJCQiIz8hIiJJIlJHRiQ= If we evaluate the position vector at time t = 4.0, we will have the position at that time: PkkjUjJHNiItSSVzdWJzRyUqcHJvdGVjdGVkRzYkL0kidEdGJCQiI1MhIiJJIlJHRiQ= If we evaluate the velocity vector at time t = 2, we will have the velocity at that time: PkkjVjFHNiItSSVzdWJzRyUqcHJvdGVjdGVkRzYkL0kidEdGJCQiIz8hIiJJIlZHRiQ= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= We plot the curve represented by R along with the velocity vector evaluated at t =1 , V1. We plot so that the initial point (tail) of the vector is at R1, the position at time t=1. QyQ+SSdQbG90VjFHNiItSStQbG90VmVjdG9yR0YlNiVJI1IxR0YlSSNWMUdGJS9JJmNvbG9yR0YlSSZibGFja0dGJSEiIg== LUkoZGlzcGxheUc2IjYkPCRJJlBsb3RSR0YkSSdQbG90VjFHRiQvSSVheGVzR0YkSSZCT1hFREdGJA== If we evaluate the velocity vector at time t = 4.0, we will have the velocity at that time: PkkjVjJHNiItSSVzdWJzRyUqcHJvdGVjdGVkRzYkL0kidEdGJCQiI1MhIiJJIlZHRiQ= Add this to the plot: QyQ+SSdQbG90VjJHNiItSStQbG90VmVjdG9yR0YlNiVJI1IyR0YlSSNWMkdGJS9JJmNvbG9yR0YlSSZibGFja0dGJSEiIg== LUkoZGlzcGxheUc2IjYkPCVJJlBsb3RSR0YkSSdQbG90VjFHRiRJJ1Bsb3RWMkdGJC9JJWF4ZXNHRiRJJkJPWEVERzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0Yk

The speed at time t is the length (or norm) of the velocity vector, which we can calculate using dot product as: PkkiU0c2Ii1JJXNxcnRHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2Iy1JMGRlbGF5RG90UHJvZHVjdEc2JEYoL0krbW9kdWxlbmFtZUdGJEksVHlwZXNldHRpbmdHRic2JEkiVkdGJEYy We can see that this simplifies: 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 Thus in this particular case, the speed (length of the velocity vector is constant). While the velocity vector changes its direction along this helix, its length remains constant

The acceleration vector a for this helix is given by a(t) = r''(t) = v'(t). PkkiQUc2Ii1JJWRpZmZHJSpwcm90ZWN0ZWRHNiRJIlZHRiRJInRHRiQ= Evaluate the acceleration at time t = 2, and time t = 4: PkkjQTFHNiItSSVzdWJzRyUqcHJvdGVjdGVkRzYkL0kidEdGJCQiIz8hIiJJIkFHRiQ= PkkjQTJHNiItSSVzdWJzRyUqcHJvdGVjdGVkRzYkL0kidEdGJCQiI1MhIiJJIkFHRiQ= We can plot the acceleration vectors (blue) and velocity vectors( black) at times t = 2 and t = 4 along with the the curve: QyQ+SSdQbG90QTFHNiItSStQbG90VmVjdG9yR0YlNiVJI1IxR0YlSSNBMUdGJS9JJmNvbG9yR0YlSSVibHVlR0YlISIi QyQ+SSdQbG90QTJHNiItSStQbG90VmVjdG9yR0YlNiVJI1IyR0YlSSNBMkdGJS9JJmNvbG9yR0YlSSVibHVlR0YlISIi LUkoZGlzcGxheUc2IjYoSSdQbG90QTFHRiRJJ1Bsb3RBMkdGJEknUGxvdFYxR0YkSSdQbG90VjJHRiRJJlBsb3RSR0YkL0klYXhlc0dGJEkmQk9YRURHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ= As expected, in both cases, the acceleration vector is "pointing in" towards the center of the helix. Note that the length of the acceleration vector is a constant 4 in this case for all values of t -- While the acceleration vector is changing direction it never changes its length. Also note that the velocity vector is perpendicular to the acceleration vector at time t=2. We can verify this by taking their dot product at this point. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2J0YrLUYjNictRiw2JVEjVjFGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYtUSIuRicvRjpRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZELyUpc3RyZXRjaHlHRkQvJSpzeW1tZXRyaWNHRkQvJShsYXJnZW9wR0ZELyUubW92YWJsZWxpbWl0c0dGRC8lJ2FjY2VudEdGRC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRlMtRiw2JVEjQTFGJ0Y2RjkvJStleGVjdXRhYmxlR0ZERkBGK0ZZRkBGK0ZZRkA= Is this always true for this particular curve, no matter what the value of t? -- Yes as we see below: LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2J0YrLUYjNictRiw2JVEiVkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIi5GJy9GOlEnbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkQvJSlzdHJldGNoeUdGRC8lKnN5bW1ldHJpY0dGRC8lKGxhcmdlb3BHRkQvJS5tb3ZhYmxlbGltaXRzR0ZELyUnYWNjZW50R0ZELyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGUy1GLDYlUSJBRidGNkY5LyUrZXhlY3V0YWJsZUdGREZARitGWUZARitGWUZA Why should we know this without caculating the dot product explicitly? -- Since length of V is _____________________________

As an second example consider the curve representing the position vector r(t) = 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 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=PkkiUkc2Ii1JL1Bvc2l0aW9uVmVjdG9yR0YkNiM3JS1JIipHNiQlKnByb3RlY3RlZEcvSSttb2R1bGVuYW1lR0YkSS9WZWN0b3JDYWxjdWx1c0c2JEYsSShfc3lzbGliR0YkNiQtSSIrR0YrNiQiIiUtSSRzaW5HRjA2I0kidEdGJC1JJGNvc0dGMDYjLUYqNiQiIiRGOi1GKjYkRjMtRjhGPS1GPEY5 An plot of this curve from t=0 to 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 is given by LUkzUGxvdFBvc2l0aW9uVmVjdG9yRzYiNiVJIlJHRiQvSSJ0R0YkOyIiIS1JIipHNiQlKnByb3RlY3RlZEcvSSttb2R1bGVuYW1lR0YkSS9WZWN0b3JDYWxjdWx1c0c2JEYuSShfc3lzbGliR0YkNiQiIiNJI1BpR0YuL0ktY3VydmVvcHRpb25zR0YkNyMvSSpudW1wb2ludHNHRiQiJCsi If we evaluate this position vector at time t = 2, we will have the position at that time: PkkjUjFHNiItSSZldmFsZkclKnByb3RlY3RlZEc2Iy1JJXN1YnNHRic2JC9JInRHRiQiIiNJIlJHRiQ= The velocity vector function is found by differentiating the position vector with respect to t:. PkkiVkc2Ii1JJWRpZmZHJSpwcm90ZWN0ZWRHNiRJIlJHRiRJInRHRiQ= If we evaluate this at time t = 2, we will have the velocity at that time: PkkjVjFHNiItSSZldmFsZkclKnByb3RlY3RlZEc2Iy1JJXN1YnNHRic2JC9JInRHRiQiIiNJIlZHRiQ= Plotting: velocity at time t =2 with the curve: QyQ+SS1QbG90VmVsb2NpdHlHNiItSStQbG90VmVjdG9yR0YlNidJI1IxR0YlSSNWMUdGJS9JJndpZHRoR0YlNyQkIiIiISIjSSlyZWxhdGl2ZUdGJS9JJmNvbG9yR0YlSSRyZWRHRiUvSSp0aGlja25lc3NHRiUiIiMhIiI= QyQ+SSpQbG90Q3VydmVHNiItSTNQbG90UG9zaXRpb25WZWN0b3JHRiU2JEkiUkdGJS9JInRHRiU7IiIhLUkiKkc2JCUqcHJvdGVjdGVkRy9JK21vZHVsZW5hbWVHRiVJL1ZlY3RvckNhbGN1bHVzRzYkRjFJKF9zeXNsaWJHRiU2JCIiI0kjUGlHRjEhIiI= LUkoZGlzcGxheUc2IjYkPCRJKlBsb3RDdXJ2ZUdGJEktUGxvdFZlbG9jaXR5R0YkL0klYXhlc0dGJEkmYm94ZWRHRiQ= The speed at time t is the length (or norm) of the velocity vector, which we can calculate as LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2JkYrLUkmbXNxcnRHRiQ2JC1GIzYnLUYsNiVRIlZGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYtUSIuRicvRj1RJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZHLyUpc3RyZXRjaHlHRkcvJSpzeW1tZXRyaWNHRkcvJShsYXJnZW9wR0ZHLyUubW92YWJsZWxpbWl0c0dGRy8lJ2FjY2VudEdGRy8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRlZGNi8lK2JhY2tncm91bmRHUS5bMjU1LDI1NSwyNTVdRidGQy8lK2JhY2tncm91bmRHRmVuRllGQ0YrRllGQw== LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbW9HRiQ2LVEiLkYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZDLyUrYmFja2dyb3VuZEdRLlsyNTUsMjU1LDI1NV1GJ0Yv PkkiU0c2Ii1JJXNxcnRHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2Iy1JMGRlbGF5RG90UHJvZHVjdEc2JEYoL0krbW9kdWxlbmFtZUdGJEksVHlwZXNldHRpbmdHRic2JEkiVkdGJEYy PkkiU0c2Ii1JKXNpbXBsaWZ5R0YkNiNGIw== For what value of t would the speed be the fastest? The slowest? Speeds at various times: QyQtSSZldmFsZkclKnByb3RlY3RlZEc2Iy1JJXN1YnNHRiU2JC9JInRHNiIiIiFJIlNHRiwiIiI=QyQtSSZldmFsZkclKnByb3RlY3RlZEc2Iy1JJXN1YnNHRiU2JC9JInRHNiItSSIqRzYkRiUvSSttb2R1bGVuYW1lR0YsSS9WZWN0b3JDYWxjdWx1c0c2JEYlSShfc3lzbGliR0YsNiRJI1BpR0YlIyIiIiIiI0kiU0dGLEY4LUkmZXZhbGZHJSpwcm90ZWN0ZWRHNiMtSSVzdWJzR0YkNiQvSSJ0RzYiLUkiLUc2JEYkL0krbW9kdWxlbmFtZUdGK0kvVmVjdG9yQ2FsY3VsdXNHNiRGJEkoX3N5c2xpYkdGKzYjLUkiKkdGLjYkSSNQaUdGJCMiIiIiIiNJIlNHRis=

Theorem 12.3 tells us that if we neglect air resistance and assume gravitational constant g, the path of a projectile launched from an initial height h with initial speed LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEidkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUkjbW5HRiQ2JFEiMEYnL0Y2USdub3JtYWxGJ0Y+LyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGPg== and initial angle of elevation LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEoJnRoZXRhO0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRidGMg== is described by the vector function r(t) = 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 i + [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] j Use this to determine the vector valued- function for Exercise 33 and graph with Maple. LUkiKkc2JCUqcHJvdGVjdGVkRy9JK21vZHVsZW5hbWVHNiJJL1ZlY3RvckNhbGN1bHVzRzYkRiVJKF9zeXNsaWJHRig2JC1GIzYkIiQrIiIlIUcmIyIiIiIlK08= PkkjdjBHNiItSSIqRzYkJSpwcm90ZWN0ZWRHL0krbW9kdWxlbmFtZUdGJEkvVmVjdG9yQ2FsY3VsdXNHNiRGKEkoX3N5c2xpYkdGJDYkIiRTJSMiIiIiIiQ= PkkiaEc2IiIiJA== Qyo+SSd0aGV0YTFHNiItSSIqRzYkJSpwcm90ZWN0ZWRHL0krbW9kdWxlbmFtZUdGJUkvVmVjdG9yQ2FsY3VsdXNHNiRGKUkoX3N5c2xpYkdGJTYkLUYnNiQiIzVJI1BpR0YpIyIiIiIkIT1GNT5JJ3RoZXRhMkdGJS1GJzYkLUYnNiQiIzpGM0Y0RjU+SSd0aGV0YTNHRiUtRic2JC1GJzYkIiM/RjNGNEY1PkkndGhldGE0R0YlLUYnNiQtRic2JCIjREYzRjRGNQ== QyQ+SSZGZW5jZUc2Ii1JJWxpbmVHRiU2JjckIiQrJSIiITckRioiIzUvSSZjb2xvckdGJUkmYmxhY2tHRiUvSSp0aGlja25lc3NHRiUiIiUhIiI= QyU+SSNSMUc2Ii1JL1Bvc2l0aW9uVmVjdG9yR0YlNiM3JC1JIipHNiQlKnByb3RlY3RlZEcvSSttb2R1bGVuYW1lR0YlSS9WZWN0b3JDYWxjdWx1c0c2JEYtSShfc3lzbGliR0YlNiQtRis2JEkjdjBHRiUtSSRjb3NHRjE2I0kndGhldGExR0YlSSJ0R0YlLUkiK0dGLDYkLUY9NiRJImhHRiUtRis2JC1GKzYkRjYtSSRzaW5HRjFGOUY7LUkiLUdGLDYjLUYrNiQiIzsqJEY7IiIjIiIiLUkmZXZhbGZHRi02I0Yk QyU+SSNSMkc2Ii1JL1Bvc2l0aW9uVmVjdG9yR0YlNiM3JC1JIipHNiQlKnByb3RlY3RlZEcvSSttb2R1bGVuYW1lR0YlSS9WZWN0b3JDYWxjdWx1c0c2JEYtSShfc3lzbGliR0YlNiQtRis2JEkjdjBHRiUtSSRjb3NHRjE2I0kndGhldGEyR0YlSSJ0R0YlLUkiK0dGLDYkLUY9NiRJImhHRiUtRis2JC1GKzYkRjYtSSRzaW5HRjFGOUY7LUkiLUdGLDYjLUYrNiQiIzsqJEY7IiIjIiIiLUkmZXZhbGZHRi02I0Yk QyU+SSNSM0c2Ii1JL1Bvc2l0aW9uVmVjdG9yR0YlNiM3JC1JIipHNiQlKnByb3RlY3RlZEcvSSttb2R1bGVuYW1lR0YlSS9WZWN0b3JDYWxjdWx1c0c2JEYtSShfc3lzbGliR0YlNiQtRis2JEkjdjBHRiUtSSRjb3NHRjE2I0kndGhldGEzR0YlSSJ0R0YlLUkiK0dGLDYkLUY9NiRJImhHRiUtRis2JC1GKzYkRjYtSSRzaW5HRjFGOUY7LUkiLUdGLDYjLUYrNiQiIzsqJEY7IiIjIiIiLUkmZXZhbGZHRi02I0Yk QyU+SSNSNEc2Ii1JL1Bvc2l0aW9uVmVjdG9yR0YlNiM3JC1JIipHNiQlKnByb3RlY3RlZEcvSSttb2R1bGVuYW1lR0YlSS9WZWN0b3JDYWxjdWx1c0c2JEYtSShfc3lzbGliR0YlNiQtRis2JEkjdjBHRiUtSSRjb3NHRjE2I0kndGhldGE0R0YlSSJ0R0YlLUkiK0dGLDYkLUY9NiRJImhHRiUtRis2JC1GKzYkRjYtSSRzaW5HRjFGOUY7LUkiLUdGLDYjLUYrNiQiIzsqJEY7IiIjIiIiLUkmZXZhbGZHRi02I0Yk QyQ+SSdQbG90UjFHNiItSTNQbG90UG9zaXRpb25WZWN0b3JHRiU2JEkjUjFHRiUvSSJ0R0YlOyIiISIiJSEiIg== QyQ+SSdQbG90UjJHNiItSTNQbG90UG9zaXRpb25WZWN0b3JHRiU2JEkjUjJHRiUvSSJ0R0YlOyIiISIiJSEiIg== QyQ+SSdQbG90UjNHNiItSTNQbG90UG9zaXRpb25WZWN0b3JHRiU2JEkjUjNHRiUvSSJ0R0YlOyIiISIiJSEiIg== QyQ+SSdQbG90UjRHNiItSTNQbG90UG9zaXRpb25WZWN0b3JHRiU2JEkjUjRHRiUvSSJ0R0YlOyIiISIiJSEiIg== LUkoZGlzcGxheUc2IjYnSSdQbG90UjFHRiRJJ1Bsb3RSMkdGJEknUGxvdFIzR0YkSSdQbG90UjRHRiRJJkZlbmNlR0Yk JSFH