# Thread: 2 tough questions i am stuck on

1. ## 2 tough questions i am stuck on

Hi there i need help with these two questions before my exam next week, if anyone can help that would be great:

firstly this one:
A moving particle at time t ∈ [0, 10] (seconds) has position vector in metres from the origin (0, 0, 0) given by the vector function r(t) = (10
− t)i + (t^2 − 10t)j + sin tk.

i. Describe the path of the particle, as seen from above (the positive k-direction), and also describe it in three dimensions.

ii. Find the curvature of the path, at t = 2π ≈ 6.28 seconds.

iii. Find the angle between the path (at start and end-points) and the k-direction.

And this one (obviously sketching might be tough)

A particle’s path, in two dimensions, is described by its position vector (in metres and time t ∈ [1, 2] seconds) relative to point (0, 0, 0) by r(t) = (2t + 1)i + (4 − t^2 )j.

i. Sketch the path of the particle.

ii. Find the value of t* at which the particle has greatest distance from (0, 0, 0).

iii. Show that at position r(t* ), the particles velocity is not perpendicular to r(t*).

ii)
The distance between your particle and the origin |r| and its square $\displaystyle |r|^2$ will be minimum at the same time t*

$\displaystyle |r|^2 = (2t+1)^2+(4-t^2)^2$

I multiplied this out to get $\displaystyle t^4 + ...$

Differentiate the result and set equal to zero to get $\displaystyle 4t^3 + ... = 0$

Find the three roots of this equation (one is t = an integer)

Ignore the negative root as negative time is silly.

Find |r| for each of the other two roots, one of these is t*.

iii) Find r(t*) from part ii)

Find v(t) = 2i - 2tj by differentiating r(t)

Find the dot product of v(t*) and r(t*). By my calculation the answer is zero proving that they are perpendicular. Could the question be wrong?

3. Originally Posted by jimmy
Hi there i need help with these two questions before my exam next week, if anyone can help that would be great:

firstly this one:
A moving particle at time t ∈ [0, 10] (seconds) has position vector in metres from the origin (0, 0, 0) given by the vector function r(t) = (10
− t)i + (t^2 − 10t)j + sin tk.

i. Describe the path of the particle, as seen from above (the positive k-direction), and also describe it in three dimensions.

ii. Find the curvature of the path, at t = 2π ≈ 6.28 seconds.

iii. Find the angle between the path (at start and end-points) and the k-direction.

And this one (obviously sketching might be tough)

A particle’s path, in two dimensions, is described by its position vector (in metres and time t ∈ [1, 2] seconds) relative to point (0, 0, 0) by r(t) = (2t + 1)i + (4 − t^2 )j.

i. Sketch the path of the particle.

ii. Find the value of t* at which the particle has greatest distance from (0, 0, 0).

iii. Show that at position r(t* ), the particles velocity is not perpendicular to r(t*).
The first question has been asked (by another member) and replied to in this thread: http://www.mathhelpforum.com/math-he...tml#post158518