A particle moves along the $x$-axis and has its displacement $x$ varying with time $t$ according to the…

A particle moves along the $x$-axis and has its displacement $x$ varying with time $t$ according to the equation $\mathrm{x}=\mathrm{c}_0\left(\mathrm{t}^2-2\right)+\mathrm{c}(\mathrm{t}-2)^2$ where $c_0$ and $c$ are constants of appropriate dimensions. Then, which of the following statements is correct?
  1. the acceleration of the particle is $2 \mathrm{c}_0$
  2. the acceleration of the particle is $2\mathrm{~c}$
  3. the initial velocity of the particle is $4\mathrm{~c}$
  4. the acceleration of the particle is $2\left(c+c_0\right)$

Solution

$\begin{aligned} & \mathrm{v}=\frac{\mathrm{dx}}{\mathrm{dt}}=2 \mathrm{tC}_0+2 \mathrm{C}(\mathrm{t}-2) \\ & \mathrm{a}=\frac{\mathrm{dv}}{\mathrm{dt}}=2 \mathrm{C}_0+2 \mathrm{C}\end{aligned}$

Asked in: JEE Main 2025 (03 Apr Shift 2)

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