Question

calculus help...plz explain and answer question...i totally dont know what they are...studying on my own

Answer 1

http://assets.openstudy.com/updates/attachments/517d65e4e4b0be6b54ab31bc-best_mathematician-1367172592503-3.png

Answer 2

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...

Answer 3

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)

Answer 4

(t-sint)i = (1-cost)i = (sint)i
(1-cost)j = (sint)j = (cost)j
f(x) f'(x) f''(x)

Answer 5

now??

Answer 6

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)

Answer 7

what is dot product

Answer 8

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)

Answer 9

ohk got it...so how to get minimum nd maximum values

Answer 10

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

Answer 11

agree

Answer 12

** 2(1 - (-1)) or 0 and 4

Answer 13

plz carry on

Answer 14

a= < sin t , cos t>
can you find its dot product (that gives you |a|^2 )

Answer 15

yes it is 1

Answer 16

sin^2t+cos^2t = 1

Answer 17

1^2 = 1

Answer 18

am i on track?

Answer 19

yes

Answer 20

ok so min and max for velocity is 0 and 4 repectively
and min and max for acceleration is 1 and 1...correct?

Answer 21

the magnitude of the acceleration is always 1, but its direction is changing.

Answer 22

oh ya but we do not need to care about direction...right?

Answer 23

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.

Answer 24

ok so how to do rest of the part

Answer 25

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.

Answer 26

ok so what is the vector function that models the behavior