What do you need to answer this question? Normal force? Lagrange's equation with a constraint? Not at all! It turns out that you can use asymptotic behavior of the answer.
First, you can eliminate choice (A) and (B) since the acceleration of the particle is not constant (otherwise, it just same as free fall or no acceleration).
Then, imagine when the particle goes very far. The tangent direction get closer and closer to y-direction. That is as if there is no track at all. The answer should converges to free fall acceleration,
, as
goes to infinity. The only answer that has this limit is (D).
The correct answer is (D). 39 of 100 people got it correctly.
To solve this question properly. One must project the gravitational force into tangential direction. (Note that the normal force due to the track is always normal to the motion). One can find the direction of tangential direction at point
as
In this case, it equals to
Let's call this vector
The projection of gravitional force
on the direction of vector
is
The norm of this tangential component equals to
The tangential acceleration can be found by just dividing the mass
out.
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