# Motion of a particle given position vector.

## Homework Statement

A position vector of a particle at a time t is r=icost +jsint +kt ; show the speed and the magnitude of the acceleration is constant. Describe the motion.

v = dr/dt
a = dv/dt

## The Attempt at a Solution

Could someone let me know if I am doing this correctly?

I derived the position to find the velocity:

v = dr/dt = -isint +jcost + 1k

Then derived the velocity :

a = dv/dt = -icost -jsint

Then found the magnitude of the acceleration:

mag(a) = sqrt ( cos^2(t) + sin^2(t)) = 1 , which is constant.

Motion- increasing oscillation? How would I show this?

Thanks!

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Don't forget to find the speed (magnitude of the velocity).

Think about just the two-dimensional $x,y$ motion. What kind of motion would that be? Then notice that the $z$ component just linearly increases in one direction. What kind of shape will be created this way? And how will your particle move along that three dimensional shape?

Don't forget to find the speed (magnitude of the velocity).

Think about just the two-dimensional $x,y$ motion. What kind of motion would that be? Then notice that the $z$ component just linearly increases in one direction. What kind of shape will be created this way? And how will your particle move along that three dimensional shape?
I took the magnitude of the velocity and got sqrt( sin^2 + cos^2 +1) so sqrt(2) , so it would also be constant.
Would the particle just be moving around a circle? How could I prove this?

Thanks!

Well, yes for the two-dimensional case it would be circular motion. If you don't recognize the form, try picking various values of $t$ and plotting them on a two dimensional graph to see it.

Is it supposed to look like a spring? And was I correct about the velocity?

Thank you for the help.

Exactly, it's the shape of a helix. The motion is circular, but it's traveling upwards along the surface of a cylinder with time.

And yes, you were right about the speed.

Awesome! Thanks!