I encountered a problem where it required me to calculate the "acceleration" vector's direction at a point given the path the particle travelled. The path is given as:
We are also given that the particle speeds up from A to B and after B, it slows down until C. Given this information, we need to find where does the acceleration vector point at the point E.
I tried 2 ways and still neither matched with the given answer. And it doesn't make sense to me.
The answer is given as:
At point E, the objects speed is instantaneously not changing; it's speed is maximum at this point, so it's derivative is zero. There is therefore no parallel component of $\vec{a}$, and the acceleration is perpendicular to it's motion.".
- So at point E, as the speed of the object is maximum and at the next instant, the speed has decreased and also the velocity vector's direction has changed. In order to calculate the acceleration, I substract the two vectors (final-initial) and then the direction of that is the direction of the acceleration. Here is the working for it.
Here, the acceleration vector is not completely in the direction of the normal, it is a bit shifted to the left due to the fact that the object is slowing down.
- This is assuming a linear first order approximation at the point E as it is a point of extrema. This gives that the acceleration vector is in the backward direction at the point E. Here is the working for this.
Both answers are different and yet I can't come to understand the answer given in the book. Where have I gone wrong? This question involves dealing more with vectors even though the question consists topics of physics, that is why I chose to post it here.