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

$x=\mathrm{c}_0\left(\mathrm{t}^2-2\right)+\mathrm{c}(\mathrm{t}-2)^2$

where $\mathrm{c}_0$ and c are constants of appropriate dimensions.

Then, which of the following statements is correct?

<p>To determine the acceleration of the particle, we start by differentiating the displacement function with respect to time $ t $.</p> <p>The displacement of the particle is given by:</p> <p>$ x = c_0(t^2 - 2) + c(t - 2)^2 $</p> <p>First, compute the velocity $ v $ by differentiating $ x $ with respect to $ t $:</p> <p>$ v = \frac{dx}{dt} = \frac{d}{dt}[c_0(t^2 - 2) + c(t - 2)^2] $</p> <p>This gives:</p> <p>$ v = 2c_0 t + 2c(t - 2) $</p> <p>Next, determine the acceleration $ a $ by differentiating the velocity function with respect to time $ t $:</p> <p>$ a = \frac{dv}{dt} = \frac{d}{dt}[2c_0 t + 2c(t - 2)] $</p> <p>Calculating this gives:</p> <p>$ a = 2c_0 + 2c $</p> <p>Hence, the acceleration of the particle is $ 2c_0 + 2c $.</p>