General Question about curved motion

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The discussion centers on determining when the speed of an object in curved motion is increasing. The user initially solved parts of the problem using the dot product and identified when the speed is constant. They questioned whether speed increases when the parallel acceleration multiplied by cos(theta) equals zero. After some analysis, they confirmed their understanding and resolved the issue. The key takeaway is that the user successfully figured out the conditions for increasing speed in curved motion.
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Homework Statement


When if ever is the speed of the object increasing?

Homework Equations


Dot product result, cos(theta) acceleration perpendicular, acceleration parralel

The Attempt at a Solution



I solved the problems first three parts and found when the objects speed is constant using the dot product, now i am trying to figure out how to find if it is increasing. Is it when a(parrelel)*cos(theta) = 0? i have my result from the dot product as 2t^3+6t^2-8t. i set it equal to 0 to find when speed is constant and got t=-4,t=1,t=0. Now i need to find speed increasing and am not sure exactly
 
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I FIGURED IT OUT!!
 

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