# particle/curvature question

• Jun 16th 2008, 01:25 PM
katie
particle/curvature question
Hi guys can anyone help me out with this?

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.

Thanks
• Jun 16th 2008, 08:59 PM
mr fantastic
Quote:

Originally Posted by katie
Hi guys can anyone help me out with this?

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.

Mr F says:

x = 10 - t => t = 10 - x .... (1)

y = t^2 - 10t .... (2)

Substitute (1) into (2).

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

Mr F says: Substitute into the formula.

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

Mr F says: Use the dot product.

Thanks

..
• Jun 16th 2008, 09:12 PM
jimmy
when you say substitute into the formula is that just into the r(t) formula?

and the dot product, i am not sure what the other vector is to be using the dot product with?

thanks
• Jun 17th 2008, 02:07 AM
mr fantastic
Quote:

Originally Posted by jimmy
when you say substitute into the formula is that just into the r(t) formula?

and the dot product, i am not sure what the other vector is to be using the dot product with?

thanks

The curvature formula.

Direction of motion given by dr/dt. Evaluate at t = 0 and t = 10. Consider the dot product of each with k ....