Find the pebble's position, velocity, and acceleration

AI Thread Summary
The discussion focuses on determining the position, velocity, and acceleration of a pebble lodged in a tire rolling without slipping. The velocity is expressed as a combination of the tire's constant speed and angular motion. To find the acceleration, the proposed formula incorporates both radial and tangential components. The position of the pebble over time can be calculated by integrating the velocity function. Cartesian coordinates are recommended for clarity, necessitating transformations based on the tire's movement.
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A tire rolls in a straight line without slipping. Its center moves with constant speed V. A small pebble lodged in the read of the tire touches the road at t = 0. Find the pebble's position, velocity, and acceleration as functions of time.

So \bold{v} = \dot{r} \bold{\hat{r}} + r \theta \bold{\hat{\theta}}.

Would it just be \bold{v} = V \bold{\hat{r}} + Vt \omega \bold{\hat{\theta}} and \bold{a} = -Vt \omega^{2} \bold{\hat{r}} + 2V \omega \bold{\hat{\theta}}?

Then to find the position as a function of time, integrate the velocity?

Thanks
 
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You need to do this in cartesian coords, because the center keeps moving.
 
So use the transformations x = r \cos \theta, y = r \sin \theta?
 
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