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r is a position vector... the position of a point that changes with time. it has components in the x direction (the "ith" component) and y direction the velocity of the point is dr/dt (derivative with respect to time) you take the derivative of each component separately.... you get a new vector that represents the velocity in the x and y dimensions...
it is not obvious, but they mean v is the velocity vector and a is the acceleration vector (2nd derivative of r, and the 1st derivative of v)
(t-sint)i = (1-cost)i = (sint)i (1-cost)j = (sint)j = (cost)j f(x) f'(x) f''(x)
r= (t-sin t) i + (1-cos t) j v= dr/dt = (1-cos t) i + sin t j or v= < (1-cos t, sin t> the magnitude of v is found by doing the dot product |v|^2 = v dot v and |v| = sqrt( v dot v)
what is dot product
Khan has videos on vectors. See http://www.khanacademy.org/math/linear-algebra/vectors_and_spaces/dot_cross_products/v/vector-dot-product-and-vector-length Here is one from the physics playlist http://www.khanacademy.org/science/physics/electricity-and-magnetism/v/the-dot-product which might be better. But the dot product is the sum of the product of corresponding elements. In this case < (1-cos t, sin t> dot < (1-cos t, sin t> = (1-cos t)^2 + (sin t)^2 which simplifies to |v| = 2(1- cos t)
ohk got it...so how to get minimum nd maximum values
we know cos t has a max of +1 and a min of -1, so |v| ranges between 2(1-1) and 2(1 - (-2)) or 0 and 4
** 2(1 - (-1)) or 0 and 4
plz carry on
a= < sin t , cos t> can you find its dot product (that gives you |a|^2 )
yes it is 1
sin^2t+cos^2t = 1
1^2 = 1
am i on track?
ok so min and max for velocity is 0 and 4 repectively and min and max for acceleration is 1 and 1...correct?
the magnitude of the acceleration is always 1, but its direction is changing.
oh ya but we do not need to care about direction...right?
you might care if you were trying to analyze what is going on... but you don't need the direction if all you care about is the magnitude.
ok so how to do rest of the part
I am not sure what they want for a plot... for the smaller hoop, make the frequency twice as fast. that means in sin t and cos t replace t with 2t this will make the hoop spin twice as fast.
ok so what is the vector function that models the behavior