Velocity, Acceleration, Unit Tangent, and Principal Unit Normal Vectors Calc IV Lab 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 The following is to convince Maple that the variable t is only real-valued (not complex). QyYtSSdhc3N1bWVHNiI2IydJInRHRiVJJXJlYWxHRiUhIiItSSppbnRlcmZhY2VHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiU2Iy9JLHNob3dhc3N1bWVkR0YlIiIhRio=

Consider a position vector r(t). The velocity vector (also called the tangent vector) at point P = r(LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiMEYnL0Y2USdub3JtYWxGJy8lK2JhY2tncm91bmRHUS5bMjU1LDI1NSwyNTVdRidGPi8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnRkBGPg==) is given by v(LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiMEYnL0Y2USdub3JtYWxGJy8lK2JhY2tncm91bmRHUS5bMjU1LDI1NSwyNTVdRidGPi8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnRkBGPg==) = r'(LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiMEYnL0Y2USdub3JtYWxGJy8lK2JhY2tncm91bmRHUS5bMjU1LDI1NSwyNTVdRidGPi8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnRkBGPg==). The direction of this vector direction 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. The unit tangent vector T(t) is the normalized "velocity" vector -- that is T(t) = r'(t)/|| r'(t)||. The unit tangent vector points in the direction of motion. (It of course also gives us the direction of the tangent line to the curve represented by r(t) The principal unit normal is defined to be N(t) = T'(t)/|| T'(t)||. It is orthogonal to T(t) (the direction of motion) and points in the direction that the curve traced by r(t) is turning at time t. (That is, it points towards the concave side of the curve). The acceleration vector a(t) can be decomposed into a linear combination of T(t) and N(t): a(t) = LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRIlRGJ0YyRjUvJStiYWNrZ3JvdW5kR1EuWzI1NSwyNTUsMjU1XUYnL0Y2USdub3JtYWxGJy8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnRj1GQA==T(t) + LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRIk5GJ0YyRjUvJStiYWNrZ3JvdW5kR1EuWzI1NSwyNTUsMjU1XUYnL0Y2USdub3JtYWxGJy8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnRj1GQA==N(t), where LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRIlRGJ0YyRjUvJStiYWNrZ3JvdW5kR1EuWzI1NSwyNTUsMjU1XUYnL0Y2USdub3JtYWxGJy8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnRj1GQA=== 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 =a.N . The scalars LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRIlRGJ0YyRjUvJStiYWNrZ3JvdW5kR1EuWzI1NSwyNTUsMjU1XUYnL0Y2USdub3JtYWxGJy8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnRj1GQA== and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRIk5GJ0YyRjUvJStiYWNrZ3JvdW5kR1EuWzI1NSwyNTUsMjU1XUYnL0Y2USdub3JtYWxGJy8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnRj1GQA== are called the tangential and normal components of acceleration. The arc length of the curve is the integral of the speed over the range of t.

As an example consider a helix curve. Maple treats lists and vectors (lists are denoted with [ ] brackets whereas vectors are denoted with < >). Note that Maple uses 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 insteak of i, j, k. PkkiUkc2Ii1JJDwsPkdJKF9zeXNsaWJHRiQ2JS1JIipHNiQlKnByb3RlY3RlZEcvSSttb2R1bGVuYW1lR0YkSS9WZWN0b3JDYWxjdWx1c0c2JUYsRiwvRi5JKFN0dWRlbnRHNiRGLEYnNiQiIiUtSSRzaW5HRjM2I0kidEdGJC1GKjYkRjUtSSRjb3NHRjNGOEY5

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: 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= If we evaluate the velocity vector at time t = 4.0, we will have the velocity at that time: PkkjVjJHNiItSSVzdWJzRyUqcHJvdGVjdGVkRzYkL0kidEdGJCQiI1MhIiJJIlZHRiQ= The arrow function, which is part of the plot library, allows one to create a line segment structure for plotting. We want to plot the vector V1, positioned with its tail at R1, at time t = 2.0 along with the helix, and similarly plot V2 with its tail at R2 at time t = 4.0, in red. QyQ+SSdQbG90VjFHNiItSSZhcnJvd0dGJTYnSSNSMUdGJUkjVjFHRiUvSSZ3aWR0aEdGJTckJCIiIyEiI0kpcmVsYXRpdmVHRiUvSSZjb2xvckdGJUkkcmVkR0YlL0kqdGhpY2tuZXNzR0YlIiIkISIi QyQ+SSpQbG90Q3VydmVHNiItSStTcGFjZUN1cnZlR0YlNiRJIlJHRiUvSSJ0R0YlOyIiIS1JIipHNiQlKnByb3RlY3RlZEcvSSttb2R1bGVuYW1lR0YlSS9WZWN0b3JDYWxjdWx1c0c2JUYxRjEvRjNJKFN0dWRlbnRHNiRGMUkoX3N5c2xpYkdGJTYkIiIlSSNQaUdGMSEiIg== QyQ+SSdQbG90VjJHNiItSSZhcnJvd0dGJTYnSSNSMkdGJUkjVjJHRiUvSSZ3aWR0aEdGJTckJCIiIyEiI0kpcmVsYXRpdmVHRiUvSSZjb2xvckdGJUkkcmVkR0YlL0kqdGhpY2tuZXNzR0YlIiIkISIi QyQ+SSpQbG90Q3VydmVHNiItSStTcGFjZUN1cnZlR0YlNiRJIlJHRiUvSSJ0R0YlOyIiIS1JIipHNiQlKnByb3RlY3RlZEcvSSttb2R1bGVuYW1lR0YlSS9WZWN0b3JDYWxjdWx1c0c2JUYxRjEvRjNJKFN0dWRlbnRHNiRGMUkoX3N5c2xpYkdGJTYkIiIlSSNQaUdGMSEiIg== LUkoZGlzcGxheUc2IjYkPCVJJ1Bsb3RWMUdGJEknUGxvdFYyR0YkSSpQbG90Q3VydmVHRiQvSSVheGVzR0YkSSZCT1hFREdGJA==

The speed at time t is the length (or norm) of the velocity vector, which we can calculate as: PkkiU0c2Ii1JJXNxcnRHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2Iy1JMGRlbGF5RG90UHJvZHVjdEc2JEYoL0krbW9kdWxlbmFtZUdGJEksVHlwZXNldHRpbmdHRic2JEkiVkdGJEYy The approximate speed at time t = 2.0 is given by: LUkmZXZhbGZHJSpwcm90ZWN0ZWRHNiMtSSVzdWJzR0YkNiQvSSJ0RzYiJCIjPyEiIkkiU0dGKw== The approximate speed at time t = 4.0 is given by: LUkmZXZhbGZHJSpwcm90ZWN0ZWRHNiMtSSVzdWJzR0YkNiQvSSJ0RzYiJCIjUyEiIkkiU0dGKw== Note that the speed is the same at time t = 2 and t = 4. If you simplify the expression for the speed S, you can see that is a constant in this case -- speed = LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkmbXNxcnRHRiQ2Iy1JI21uR0YkNiRRIzE3RicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUrYmFja2dyb3VuZEdRLlsyNTUsMjU1LDI1NV1GJ0Yy. QyUtSSlzaW1wbGlmeUc2IjYjSSJTR0YlIiIiLUkmZXZhbGZHJSpwcm90ZWN0ZWRHNiMtSSVzcXJ0RzYkRitJKF9zeXNsaWJHRiU2IyIjPA==

The acceleration vector a for this helix is given by a(t) = r''(t) = v'(t). PkkiQUc2Ii1JJWRpZmZHJSpwcm90ZWN0ZWRHNiRJIlZHRiRJInRHRiQ= To evaluate the acceleration at time t = 2, and time t = 4: QyQ+SSNBMUc2Ii1JJXN1YnNHJSpwcm90ZWN0ZWRHNiQvSSJ0R0YlJCIjPyEiIkkiQUdGJSIiIg==PkkjQTJHNiItSSVzdWJzRyUqcHJvdGVjdGVkRzYkL0kidEdGJCQiI1MhIiJJIkFHRiQ= We can plot the acceleration vectors (blue) and velocity vectors( red) at times t =2 and t = 4 along with the the curve: QyQ+SSdQbG90QTFHNiItSSZhcnJvd0dGJTYnSSNSMUdGJUkjQTFHRiUvSSZ3aWR0aEdGJTckJCIiIyEiI0kpcmVsYXRpdmVHRiUvSSZjb2xvckdGJUklYmx1ZUdGJS9JKnRoaWNrbmVzc0dGJSIiJCIiIg==QyQ+SSdQbG90QTJHNiItSSZhcnJvd0dGJTYnSSNSMkdGJUkjQTJHRiUvSSZ3aWR0aEdGJTckJCIiIyEiI0kpcmVsYXRpdmVHRiUvSSZjb2xvckdGJUklYmx1ZUdGJS9JKnRoaWNrbmVzc0dGJSIiJCIiIg==LUkoZGlzcGxheUc2IjYoSSdQbG90QTFHRiRJJ1Bsb3RWMUdGJEknUGxvdEEyR0YkSSdQbG90VjJHRiRJKlBsb3RDdXJ2ZUdGJC9JJWF4ZXNHRiRJJkJPWEVERzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0Yk 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. LUkwZGVsYXlEb3RQcm9kdWN0RzYkJSpwcm90ZWN0ZWRHL0krbW9kdWxlbmFtZUc2IkksVHlwZXNldHRpbmdHNiRGJUkoX3N5c2xpYkdGKDYkSSNWMUdGKEkjQTFHRig= Is this always true for this helix, 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 magnitude of V is ________________________

The unit tangent vector is the normalized "velocity" vector. Note that it is a function of t, not a constant. PkkiVEc2Ii1JIipHNiQlKnByb3RlY3RlZEcvSSttb2R1bGVuYW1lR0YkSS9WZWN0b3JDYWxjdWx1c0c2JUYoRigvRipJKFN0dWRlbnRHNiRGKEkoX3N5c2xpYkdGJDYkSSJWR0YkKiQtSSVzcXJ0R0YvNiMtSTBkZWxheURvdFByb2R1Y3RHNiRGKC9GKkksVHlwZXNldHRpbmdHRi82JEYyRjIhIiI= PkkiVEc2Ii1JKXNpbXBsaWZ5RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiNGIw== The principal unit normal vector is obtained by differentiating the unit tangent vector and then normalizing its length to one. QyQ+SSNEVEc2Ii1JJWRpZmZHJSpwcm90ZWN0ZWRHNiRJIlRHRiVJInRHRiUiIiI=PkkiTkc2Ii1JIipHNiQlKnByb3RlY3RlZEcvSSttb2R1bGVuYW1lR0YkSS9WZWN0b3JDYWxjdWx1c0c2JUYoRigvRipJKFN0dWRlbnRHNiRGKEkoX3N5c2xpYkdGJDYkSSNEVEdGJCokLUklc3FydEdGLzYjLUkwZGVsYXlEb3RQcm9kdWN0RzYkRigvRipJLFR5cGVzZXR0aW5nR0YvNiRGMkYyISIi The PrincipalNormal command returns the unnormalized PrincipalNormal (you would have to normalize to get the Principal Unit Normal) QyQ+SSNQTkc2Ii1JMFByaW5jaXBhbE5vcm1hbEdGJTYjSSJSR0YlIiIiPkkiTkc2Ii1JIipHNiQlKnByb3RlY3RlZEcvSSttb2R1bGVuYW1lR0YkSS9WZWN0b3JDYWxjdWx1c0c2JUYoRigvRipJKFN0dWRlbnRHNiRGKEkoX3N5c2xpYkdGJDYkSSNQTkdGJCokLUklc3FydEdGLzYjLUkwZGVsYXlEb3RQcm9kdWN0RzYkRigvRipJLFR5cGVzZXR0aW5nR0YvNiRGMkYyISIi In either case we need to tell Maple to simplify: PkkiTkc2Ii1JKXNpbXBsaWZ5RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiNGIw== The tangential and normal components of acceleration are calculated as follows: PkkjQVRHNiItSTBkZWxheURvdFByb2R1Y3RHNiQlKnByb3RlY3RlZEcvSSttb2R1bGVuYW1lR0YkSSxUeXBlc2V0dGluZ0c2JEYoSShfc3lzbGliR0YkNiRJIkFHRiRJIlRHRiQ= QyQ+SSNBTkc2Ii1JMGRlbGF5RG90UHJvZHVjdEc2JCUqcHJvdGVjdGVkRy9JK21vZHVsZW5hbWVHRiVJLFR5cGVzZXR0aW5nRzYkRilJKF9zeXNsaWJHRiU2JEkiQUdGJUkiTkdGJSIiIg==LUkpc2ltcGxpZnlHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2I0kjQU5HRic=JSFH This is as we should have expected -- the acceleration vector a(t) = LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRIlRGJ0YyRjUvJStiYWNrZ3JvdW5kR1EuWzI1NSwyNTUsMjU1XUYnL0Y2USdub3JtYWxGJy8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnRj1GQA==T(t) + LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRIk5GJ0YyRjUvJStiYWNrZ3JvdW5kR1EuWzI1NSwyNTUsMjU1XUYnL0Y2USdub3JtYWxGJy8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnRj1GQA==N(t) = 4N(t), is in the direction of the unit normal vector for all values of t ( pointing towards the center of the helix). This should make sense to us -- since the magnitude of the velocity vector is constant, all of the acceration (rate of change of the velocity) is expended in "turning".

As an second example consider the curve representing the position vector r(t) = 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 JSFHPkkiUkc2Ii1JJDwsPkdJKF9zeXNsaWJHRiQ2JS1JIipHNiQlKnByb3RlY3RlZEcvSSttb2R1bGVuYW1lR0YkSS9WZWN0b3JDYWxjdWx1c0c2JUYsRiwvRi5JKFN0dWRlbnRHNiRGLEYnNiQtSSIrR0YrNiQiIiUtSSRzaW5HRjM2I0kidEdGJC1JJGNvc0dGMzYjLUYqNiQiIiRGPC1GKjYkRjUtRjpGPy1GPkY7 An plot of this curve from t=0 to 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 is given by LUkrU3BhY2VDdXJ2ZUc2IjYkSSJSR0YkL0kidEdGJDsiIiEtSSIqRzYkJSpwcm90ZWN0ZWRHL0krbW9kdWxlbmFtZUdGJEkvVmVjdG9yQ2FsY3VsdXNHNiVGLkYuL0YwSShTdHVkZW50RzYkRi5JKF9zeXNsaWJHRiQ2JCIiI0kjUGlHRi4= 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+SS1QbG90VmVsb2NpdHlHNiItSSZhcnJvd0dGJTYnSSNSMUdGJUkjVjFHRiUvSSZ3aWR0aEdGJTckJCIiIiEiI0kpcmVsYXRpdmVHRiUvSSZjb2xvckdGJUkkcmVkR0YlL0kqdGhpY2tuZXNzR0YlIiIjISIi QyQ+SSpQbG90Q3VydmVHNiItSStTcGFjZUN1cnZlR0YlNiRJIlJHRiUvSSJ0R0YlOyIiIS1JIipHNiQlKnByb3RlY3RlZEcvSSttb2R1bGVuYW1lR0YlSS9WZWN0b3JDYWxjdWx1c0c2JUYxRjEvRjNJKFN0dWRlbnRHNiRGMUkoX3N5c2xpYkdGJTYkIiIjSSNQaUdGMSEiIg== LUkoZGlzcGxheUc2IjYkPCRJKlBsb3RDdXJ2ZUdGJEktUGxvdFZlbG9jaXR5R0YkL0klYXhlc0dGJEkmYm94ZWRHRiQ= The speed at time t is the length (or norm) of the velocity vector, which we can calculate as LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2JkYrLUkmbXNxcnRHRiQ2JC1GIzYnLUYsNiVRIlZGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYtUSIuRicvRj1RJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZHLyUpc3RyZXRjaHlHRkcvJSpzeW1tZXRyaWNHRkcvJShsYXJnZW9wR0ZHLyUubW92YWJsZWxpbWl0c0dGRy8lJ2FjY2VudEdGRy8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRlZGNi8lK2JhY2tncm91bmRHUS5bMjU1LDI1NSwyNTVdRidGQy8lK2JhY2tncm91bmRHRmVuRllGQ0YrRllGQw== LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbW9HRiQ2LVEiLkYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZDLyUrYmFja2dyb3VuZEdRLlsyNTUsMjU1LDI1NV1GJ0Yv PkkiU0c2Ii1JJXNxcnRHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2Iy1JMGRlbGF5RG90UHJvZHVjdEc2JEYoL0krbW9kdWxlbmFtZUdGJEksVHlwZXNldHRpbmdHRic2JEkiVkdGJEYy PkkiU0c2Ii1JKXNpbXBsaWZ5RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiNGIw== For what value of t would the speed be the fastest? The slowest? Speeds at various times: QyQtSSZldmFsZkclKnByb3RlY3RlZEc2Iy1JJXN1YnNHRiU2JC9JInRHNiIiIiFJIlNHRiwiIiI=QyQtSSZldmFsZkclKnByb3RlY3RlZEc2Iy1JJXN1YnNHRiU2JC9JInRHNiItSSIqRzYkRiUvSSttb2R1bGVuYW1lR0YsSS9WZWN0b3JDYWxjdWx1c0c2JUYlRiUvRjFJKFN0dWRlbnRHNiRGJUkoX3N5c2xpYkdGLDYkSSNQaUdGJSMiIiIiIiNJIlNHRixGOw==LUkmZXZhbGZHJSpwcm90ZWN0ZWRHNiMtSSVzdWJzR0YkNiQvSSJ0RzYiLUkiLUc2JEYkL0krbW9kdWxlbmFtZUdGK0kvVmVjdG9yQ2FsY3VsdXNHNiVGJEYkL0YwSShTdHVkZW50RzYkRiRJKF9zeXNsaWJHRis2Iy1JIipHRi42JEkjUGlHRiQjIiIiIiIjSSJTR0Yr We can see we get a really, really, really complicated expression for the unit tangent vector and principal unit normal vector: PkkiVEc2Ii1JIipHNiQlKnByb3RlY3RlZEcvSSttb2R1bGVuYW1lR0YkSS9WZWN0b3JDYWxjdWx1c0c2JUYoRigvRipJKFN0dWRlbnRHNiRGKEkoX3N5c2xpYkdGJDYkSSJWR0YkKiQtSSVzcXJ0R0YvNiMtSTBkZWxheURvdFByb2R1Y3RHNiRGKC9GKkksVHlwZXNldHRpbmdHRi82JEYyRjIhIiI= LUkpc2ltcGxpZnlHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2I0kiVEdGJw== QyQ+SSNEVEc2Ii1JJWRpZmZHJSpwcm90ZWN0ZWRHNiRJIlRHRiVJInRHRiUiIiI=PkkiTkc2Ii1JIipHNiQlKnByb3RlY3RlZEcvSSttb2R1bGVuYW1lR0YkSS9WZWN0b3JDYWxjdWx1c0c2JUYoRigvRipJKFN0dWRlbnRHNiRGKEkoX3N5c2xpYkdGJDYkSSNEVEdGJCokLUklc3FydEdGLzYjLUkwZGVsYXlEb3RQcm9kdWN0RzYkRigvRipJLFR5cGVzZXR0aW5nR0YvNiRGMkYyISIi

Complete Exercise 58, page 864 by completing the commands below, documenting each with a text cell explaining what you are computing: 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 JSFH Plotting: T(t) amd N(t) at time t =2 with the curve: 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