Position, Velocity, Acceleration Vectors and Projectile Motion Calc IV Lab Karen Donnelly Saint Joseph's College

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEocmVzdGFydEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIjpGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1GNjYtUSJ+RidGOUY7Rj5GQEZCRkRGRkZIL0ZLUSYwLjBlbUYnL0ZORlM= QyYtSSV3aXRoRzYiNiNJJnBsb3RzRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlISIiLUYkNiNJKnBsb3R0b29sc0dGKEYr QyQtSSV3aXRoRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiNJL1ZlY3RvckNhbGN1bHVzR0YlISIi The following is to convince Maple that the variable t is only real-valued (not complex). QyYtSSdhc3N1bWVHNiI2IydJInRHRiVJJXJlYWxHRiUhIiItSSppbnRlcmZhY2VHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiU2Iy9JLHNob3dhc3N1bWVkR0YlIiIhRio= JSFHPkktX0VudkV4cGxpY2l0RzYiSSV0cnVlRyUqcHJvdGVjdGVkRw== Note that the above command "_EnvExplicit := true" makes sure that all roots of polynomials are shown as explicit solutions.

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 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiMEYnL0Y2USdub3JtYWxGJy8lK2JhY2tncm91bmRHUS5bMjU1LDI1NSwyNTVdRidGPi8lL3N1YnNjcmlwdHNoaWZ0R0Y9RkBGPg==, and its magnitude || r'(LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiMEYnL0Y2USdub3JtYWxGJy8lK2JhY2tncm91bmRHUS5bMjU1LDI1NSwyNTVdRidGPi8lL3N1YnNjcmlwdHNoaWZ0R0Y9RkBGPg==)|| is the speed of the object at time LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiMEYnL0Y2USdub3JtYWxGJy8lK2JhY2tncm91bmRHUS5bMjU1LDI1NSwyNTVdRidGPi8lL3N1YnNjcmlwdHNoaWZ0R0Y9RkBGPg==. 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. JSFH

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= 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 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2J0YrLUYjNictRiw2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GOlEnbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkQvJSlzdHJldGNoeUdGRC8lKnN5bW1ldHJpY0dGRC8lKGxhcmdlb3BHRkQvJS5tb3ZhYmxlbGltaXRzR0ZELyUnYWNjZW50R0ZELyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGUy1JI21uR0YkNiRRIzEuRidGQC8lK2JhY2tncm91bmRHUS5bMjU1LDI1NSwyNTVdRidGQEYrRlpGQEYrRlpGQA== 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: PkkiU0c2Ii1JKXNpbXBsaWZ5R0YkNiNGIw== 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 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2J0YrLUYjNictRiw2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GOlEnbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkQvJSlzdHJldGNoeUdGRC8lKnN5bW1ldHJpY0dGRC8lKGxhcmdlb3BHRkQvJS5tb3ZhYmxlbGltaXRzR0ZELyUnYWNjZW50R0ZELyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGUy1JI21uR0YkNiRRIjJGJ0ZALyUrYmFja2dyb3VuZEdRLlsyNTUsMjU1LDI1NV1GJ0ZARitGWkZARitGWkZA. We can verify this by taking their dot product at this point. LUkwZGVsYXlEb3RQcm9kdWN0RzYkJSpwcm90ZWN0ZWRHL0krbW9kdWxlbmFtZUc2IkksVHlwZXNldHRpbmdHNiRGJUkoX3N5c2xpYkdGKDYkSSNWMUdGKEkjQTFHRig= Is this always true for this particular curve, no matter what the value of t? -- Yes as we see below: LUkwZGVsYXlEb3RQcm9kdWN0RzYkJSpwcm90ZWN0ZWRHL0krbW9kdWxlbmFtZUc2IkksVHlwZXNldHRpbmdHNiRGJUkoX3N5c2xpYkdGKDYkSSJWR0YoSSJBR0Yo 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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 JSFHPkkiUkc2Ii1JL1Bvc2l0aW9uVmVjdG9yR0YkNiM3JS1JIipHNiQlKnByb3RlY3RlZEcvSSttb2R1bGVuYW1lR0YkSS9WZWN0b3JDYWxjdWx1c0c2JEYsSShfc3lzbGliR0YkNiQtSSIrR0YrNiQiIiUtSSRzaW5HRjA2I0kidEdGJC1JJGNvc0dGMDYjLUYqNiQiIiRGOi1GKjYkRjMtRjhGPS1GPEY5 An plot of this curve from t=0 to 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 is given by QyU+SSpQbG90Q3VydmVHNiItSTNQbG90UG9zaXRpb25WZWN0b3JHRiU2JUkiUkdGJS9JInRHRiU7IiIhLUkiKkc2JCUqcHJvdGVjdGVkRy9JK21vZHVsZW5hbWVHRiVJL1ZlY3RvckNhbGN1bHVzRzYkRjFJKF9zeXNsaWJHRiU2JCIiI0kjUGlHRjEvSS1jdXJ2ZW9wdGlvbnNHRiU3Iy9JKm51bXBvaW50c0dGJSIkKyIhIiJGJA== 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 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2J0YrLUYjNictRiw2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GOlEnbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkQvJSlzdHJldGNoeUdGRC8lKnN5bW1ldHJpY0dGRC8lKGxhcmdlb3BHRkQvJS5tb3ZhYmxlbGltaXRzR0ZELyUnYWNjZW50R0ZELyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGUy1JI21uR0YkNiRRIjJGJ0ZALyUrYmFja2dyb3VuZEdRLlsyNTUsMjU1LDI1NV1GJ0ZARitGWkZARitGWkZA with the curve: QyQ+SS1QbG90VmVsb2NpdHlHNiItSStQbG90VmVjdG9yR0YlNidJI1IxR0YlSSNWMUdGJS9JJndpZHRoR0YlNyQkIiIiISIjSSlyZWxhdGl2ZUdGJS9JJmNvbG9yR0YlSSRyZWRHRiUvSSp0aGlja25lc3NHRiUiIiMhIiI= QyQ+SSpQbG90Q3VydmVHNiItSTNQbG90UG9zaXRpb25WZWN0b3JHRiU2JUkiUkdGJS9JInRHRiU7IiIhLUkiKkc2JCUqcHJvdGVjdGVkRy9JK21vZHVsZW5hbWVHRiVJL1ZlY3RvckNhbGN1bHVzRzYkRjFJKF9zeXNsaWJHRiU2JCIiI0kjUGlHRjEvSS1jdXJ2ZW9wdGlvbnNHRiU3Iy9JKm51bXBvaW50c0dGJSIkKyIhIiI= LUkoZGlzcGxheUc2IjYkPCRJKlBsb3RDdXJ2ZUdGJEktUGxvdFZlbG9jaXR5R0YkL0klYXhlc0dGJEkmYm94ZWRHRiQ= The speed at time t is the length (or norm) of the velocity vector, which we can calculate as LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2JkYrLUkmbXNxcnRHRiQ2JC1GIzYnLUYsNiVRIlZGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYtUSIuRicvRj1RJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZHLyUpc3RyZXRjaHlHRkcvJSpzeW1tZXRyaWNHRkcvJShsYXJnZW9wR0ZHLyUubW92YWJsZWxpbWl0c0dGRy8lJ2FjY2VudEdGRy8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRlZGNi8lK2JhY2tncm91bmRHUS5bMjU1LDI1NSwyNTVdRidGQ0ZZRllGQ0YrRllGQw== LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbW9HRiQ2LVEiLkYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZDLyUrYmFja2dyb3VuZEdRLlsyNTUsMjU1LDI1NV1GJ0Yv PkkiU0c2Ii1JJXNxcnRHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2Iy1JMGRlbGF5RG90UHJvZHVjdEc2JEYoL0krbW9kdWxlbmFtZUdGJEksVHlwZXNldHRpbmdHRic2JEkiVkdGJEYy PkkiU0c2Ii1JKXNpbXBsaWZ5RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiNGIw== For what value of t would the speed be the fastest? The slowest? Speed at various times: QyQtSSZldmFsZkclKnByb3RlY3RlZEc2Iy1JJXN1YnNHRiU2JC9JInRHNiIiIiFJIlNHRiwiIiI=QyQtSSZldmFsZkclKnByb3RlY3RlZEc2Iy1JJXN1YnNHRiU2JC9JInRHNiItSSIqRzYkRiUvSSttb2R1bGVuYW1lR0YsSS9WZWN0b3JDYWxjdWx1c0c2JEYlSShfc3lzbGliR0YsNiRJI1BpR0YlIyIiIiIiI0kiU0dGLEY4LUkmZXZhbGZHJSpwcm90ZWN0ZWRHNiMtSSVzdWJzR0YkNiQvSSJ0RzYiLUkiLUc2JEYkL0krbW9kdWxlbmFtZUdGK0kvVmVjdG9yQ2FsY3VsdXNHNiRGJEkoX3N5c2xpYkdGKzYjLUkiKkdGLjYkSSNQaUdGJCMiIiIiIiNJIlNHRis=

This exercise is based on Larson, Exercise 33, Section 12.3 A baseball is hit from a hieght of 3 feet above the ground with an initial velocity of 100 miles per hour in the Sky Dome in Toronto, Ontario. Use the launch angles of 10, 15, 20, and 25 degrees to graph the paths of the baseball and determine the minimum angle for a home run. The SkyDome centerfield fence is 10 feet high and 400 feet from home plate. Then determine numerically the minimum angle to clear the fence for a home run. 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+LyUvc3Vic2NyaXB0c2hpZnRHRj1GPg== 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 and graph with Maple. Be sure to convert the angles to radians for Maple. Also be sure to convert the velocity to feet per second. 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 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= Numeric solution for minimum angle: Solve simultaneously the equations for 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 for the angle LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2JS1GLDYlUScmIzk1MjtGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUrYmFja2dyb3VuZEdRLlsyNTUsMjU1LDI1NV1GJ0Y3RitGOkY3. PkkiUkc2Ii1JL1Bvc2l0aW9uVmVjdG9yR0YkNiM3JC1JIipHNiQlKnByb3RlY3RlZEcvSSttb2R1bGVuYW1lR0YkSS9WZWN0b3JDYWxjdWx1c0c2JEYsSShfc3lzbGliR0YkNiQtRio2JEkjdjBHRiQtSSRjb3NHRjA2I0kmdGhldGFHRiRJInRHRiQtSSIrR0YrNiQtRjw2JEkiaEdGJC1GKjYkLUYqNiRGNS1JJHNpbkdGMEY4RjotSSItR0YrNiMtRio2JCIjOyokRjoiIiM= PkkkYW5zRzYiLUknZnNvbHZlR0YkNiU8JC8mSSJSR0YkNiMiIiIiJCslLyZGKzYjIiIjIiM1PCRJInRHRiRJJnRoZXRhR0YkL0Y2OyIiISQiIiYhIiI= LUkmZXZhbGZHJSpwcm90ZWN0ZWRHNiMtSSIqRzYkRiQvSSttb2R1bGVuYW1lRzYiSS9WZWN0b3JDYWxjdWx1c0c2JEYkSShfc3lzbGliR0YrNiQtRic2JC1JJHJoc0dGJDYjJkkkYW5zR0YrNiMiIiIiJCE9KiRJI1BpR0YkISIi JSFH

From the VectorCalculus package: PositionVector PlotVector PlotPositionVector subs fsolve: Numerically solve equations.