## Best_Mathematician 2 years ago calculus help...plz explain and answer question...i totally dont know what they are...studying on my own

1. Best_Mathematician
2. phi

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...

3. phi

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)

4. Best_Mathematician

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

5. Best_Mathematician

now??

6. phi

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)

7. Best_Mathematician

what is dot product

8. phi

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)

9. Best_Mathematician

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

10. phi

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

11. Best_Mathematician

agree

12. phi

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

13. Best_Mathematician

plz carry on

14. phi

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

15. Best_Mathematician

yes it is 1

16. Best_Mathematician

sin^2t+cos^2t = 1

17. Best_Mathematician

1^2 = 1

18. Best_Mathematician

am i on track?

19. phi

yes

20. Best_Mathematician

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

21. phi

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

22. Best_Mathematician

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

23. phi

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.

24. Best_Mathematician

ok so how to do rest of the part

25. phi

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.

26. Best_Mathematician

ok so what is the vector function that models the behavior