Question

A particle is moving along the curve $x={at}^2+bt+c$. If $ac=b^2$, then the particle would be moving with uniform

• A. rotation
• B. velocity
• C. acceleration
• D. retardation

Answer 1

The correct option is C acceleration

The position of the particle can be described by the curve whose expression is mentioned below.
$x=a{t}^2+bt+c$
On differentiating $x$ with respect to $t$, we get:

$velocity=\frac{dx}{dt}=2at+b$
On differentiating $velocity$ with respect to $t$, we get:

$Acceleration=\frac{dv}{dt}=\frac{d^{2}x}{d{t}^2}=2a$

On differentiating the given equation of the curve repeatidly till we get the constant term with respect to time, we can observe that acceleration is independent of $t$ and is equal to $2a$ which is a constant.
Hence, here the particle is moving with a uniform acceleration of magnitude of $2a$.
So, the correct option will be $\left(C\right)$.