A particle moves along the parabolic path y=a x^2 in such a...

A particle moves along the parabolic path $y=a x^{2}$ in such a way that the $x$ component of the velocity remains constant, say $c$. The acceleration of the particle is
(A) $a c \hat{k}$
(B) $2 a c^{2} \hat{j}$
(C) $2 a c^{2} \hat{k}$
(D) $a^{2} c \hat{j}$

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Key Concepts

Application of the Chain Rule

The chain rule is a key mathematical tool used to differentiate composite functions. When a coordinate is defined as a function of another variable that itself changes with time, the chain rule allows for the computation of the time derivative by linking the rate of change of the inner function with the derivative of the outer function.

Parametric Representation of Motion

This concept involves expressing the position of a particle in terms of a parameter, usually time, which allows separate analysis of each spatial component. By representing the x and y coordinates as functions of time, one can easily derive expressions for velocity and acceleration in each direction.

Velocity and Acceleration Vectors

The velocity vector is defined as the time derivative of the position vector and indicates both the speed and direction of motion. Acceleration, being the derivative of velocity with respect to time, provides insight into how the velocity changes. Analyzing these vectors component-wise is crucial, especially when one component (like the horizontal one) is constant.

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