Cal 3 problem that is eating my brain - particle along a spiral path

[houdinilogic]

Homework Statement

A particle is moving down a spiral path parameterized by: x = cos(au), y = sin(au), z = -u, where 0 \leq u \leq b (a,b real, >0). Starting from rest, the particle moves down the spiral under the influence of gravity and free from friction. Let g be the positive gravitational constant (so we're not assuming any value for g or particular units)

Find the position vector as a function of time.

Find the time it takes the particle to travel from the top of the spiral to the bottom.

Find the distance traveled by the particle.

Find the velocity of the particle when it reaches the bottom.

Homework Equations

The Attempt at a Solution

Initially, I used the vector < 0, 0, -g > for the acceleration, then integrated w/ respect to t (twice) to obtain expressions for the velocity and position in terms of t... then plugged in the expression (1/2)gt^2 for u in the parameterization to get the position vector of the particle. I feel pretty confident that this isn't correct.

I think I need to make use of the tangential acceleration, but I'm not entirely sure what to do. I think this problem is eating my brain, and would greatly appreciate any help :)

 

[tt2348]

you want to draw a free body diagram, and consider the components of gravity, one in the x direction, one in the y direction, one in the z, depending on the angle. when the ball is on the x axis, it would look like an object on an incline right?

 

[houdinilogic]

Yes, I've gotten that much. I think the part that's really giving me problems is finding a way to express the position as a vector in terms of t.

 

[tt2348]

I'm going to work on it between class today and see what I come up with . Since I'm assuming the position starts above ground, I'm going to assume z=u positive. And where does b come into play btw?

 

[tt2348]

OK I've worked on it , and this is what I've come up with.

im attaching it as a pdf file because it's too much to latex , and I don't have the time to write it all out, I apologize if the writing is too messy, but this should put you in a better direction .

oh and incase you're curious, $$\dot{x}=\frac{dx}{dt}$$ (Newtonian notation).