MAC2313 Lecture Notes - Lecture 9: Osculating Plane, Dot Product, Frenet–Serret Formulas

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plummoose320

20 Dec 2017

School

Miami Dade College

Department

Mathematics: Calculus & Pre-Calculus

Course

MAC2313

Professor

Staff

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10. 9 motion in space - velocity and acceleration notes: sterling. Provide a generalization to each of the key terms listed in this section. If you have a particle that"s moving through space, then you can get r(t) be the position vector while at t (the time). Magnitude helps determine the actual size of the vector"s displacement per unit time. Thanks to the limit, then it means that the vector vector is also classi ed as the tangent vector in this case along with the general point being on the tangent line"s general direction. A particle"s speed while it at a certain t (time) would also be known as the velocity vector"s magnitude, which can be classi ed as the following: If that is the case, then following is how the writing can be for the speed (as the general rate of change (of the given distance) while it"s respective to its time):

Related Questions

Just Need 2,3,4,5 :) Thank you so much!

Matt and his friends are enjoying an afternoon at a baseball game. A batter hits a towering homerun, and
Matt shouts, âWow, that must have been 110 feet high!â The ball was 4 feet off the ground when the batter
hit it, and the ball came off the bat traveling vertically at 80 feet per second.

a. Model the ballâs height h (in feet) at time t (in seconds) using the projectile motion model h(t) = 16t2 + v0t + h0 where v0 is the projectileâs initial vertical velocity (in feet per second) and h0 is the projectileâs initial height (in feet). Use the model to write an equation based on Mattâs claim, and then determine whether Mattâs claim is correct.

b. Did the ball reach a height of 100 feet? Explain.

c. Let hmax be the ballâs maximum height. By setting the projectile motion model equal to hmax, show
how you can find hmax using the discriminant of the quadratic formula.

d. Find the time at which the ball reached its maximum height.

Page 3 of 5 which gives us y as a function of Theoretically. the integration on the right of t= 1/ - g + alpha f(V)dv can be performed and the result could provide velocity. v,. as a function of t; Then we could perform the integration in y=y0 + V/- g + alpha f(V)dv and substitute v(t) to obtain y as a function of t. Theoretically, that is. Let's look at in example An Example Suppose the air resistance model for a particular object in free fall has where units for v area feet/second Note how f is defined in terms of the sign of v. As usual. downward has been selected as the negative direction so velocities v of an object in free fall will have negative values Thus -v will be positive and the square root makes. sense Similarly if the object is moving upward, v> 0. the value of f(v)) is negative. The object is dropped from a height: of 500 feet with no initial velocity Using g = 32, the integral models from above becomes and where we have used the definition of f(v) for negative values of v since the object is in free fall With a little hard work or the use of a CAS, the first equation can be written as Page 4 of 5 and the second as Now if we could solve the first expression for v in terms of r and substitute the result for v in the expression for y, the result would be very unwieldy to work with. Actually it is impossible to solve the first expression to get a nice closed form for v. Thanks to technology. however, these expressions can still be used to study the object's motion Choose any value of v. say v=-1. Substituting -1 for v in the equations for t and y tells us that t = 0.032 and y = 499.984 so the point (0.032, 499.984) would lie on a graph of distance versus time Similarly, setting v = -50 provides the point (1 839, 452 462) By using a wide range of values for v. we can generate the complete graph of the distance function as a function of time Fortunately, most CAS provides the capability of plotting the points (t(v), y(v)) foe a range of values of v without having to compute each point separately Such a graph appears below. One final note. It is also possible that the integrals representing t and y cannot be integrated into a closed form No problem In this case, simply use the expressions t= 1/ - g + alpha f(V)dV y=y0+ V/ - g + alpha f(V)dV Page 5 of 5 and the numerical integrators provided by a CAS to determine values of t und y Exercises In all of the exercises you will need a CAS or related tool In the example above the graph indicates the object strikes the ground some time between 6 and 7 seconds from release Determine this time to three decimal places In the case where alpha = 1/2 ,f(v) = -v, v0 = 0, y0=100 determine closed form. expressions for v and y as function. Of t. An object is thrown upward at 40 ft/sec from a height of 100 feet The air resistance can be modeled by alpha =0.001 and f(v) = - |v| v Using the trajectory model determine the speed of the object when it strike the ground, determine the time the object strikes the ground after takeoff, construct the graph of the object's height versus time.

MAC2313 Lecture Notes - Lecture 9: Osculating Plane, Dot Product, Frenet–Serret Formulas

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