A particular point moves on the parabola $y^{2} = 4 a x$ in such a way that its projection on $y$ -axis has a constant velocity. Then its projection on $x$ -axis moves with

constant velocity

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constant acceleration

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variable velocity

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variable acceleration

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Solution

The correct options are
B constant acceleration
C variable velocity
$y^{2} = 4 a x$
Differentiating with respect to t=time, we get
$2 y . \frac{d y}{d t} = 4 a . \frac{d x}{d t}$
Hence
$2 y . v_{y} = 4 a . v_{x}$
Now $v_{y}$ is constant with time.
However $v_{x}$ varies with y.
Again differentiating with respect to t, we get
$2 \frac{d y}{d t} . v_{y} + 2 y . \frac{d v_{y}}{d t} = 4 a . \frac{d v_{x}}{d t}$
$2. v_{y}^{2} + 0 = 4 a . A_{x}$
Or
$4 a A_{x} = 2 v_{y}^{2}$
Or
$A_{x} = \frac{v_{y}^{2}}{2 a}$
Hence $A_{x} = a c c e l e r a t i o n$ is constant with time.

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A particular point moves on the parabola y2=4ax in such a way that its projection on y-axis has a constant velocity. Then its projection on x-axis moves with

A particular point moves on the parabola $y^{2} = 4 a x$ in such a way that its projection on $y$ -axis has a constant velocity. Then its projection on $x$ -axis moves with