The relation between time '$t$' and distance '$x$' is $t=\alpha x^2+\beta x$, where $\alpha$ and $\beta$ are constants. The relation between acceleration $(a)$ and velocity $(v)$ is :

<p>The relationship between time ($t$) and distance ($x$) is given by $t = \alpha x^2 + \beta x$, where $\alpha$ and $\beta$ are constants. To find the relation between acceleration ($a$) and velocity ($v$), we can follow these steps:</p> <p>First, we differentiate the given equation with respect to time:</p> <p>$$\begin{aligned} & t = \alpha x^2 + \beta x \quad \text{(differentiating with respect to time)} \\ & \frac{dt}{dx} = 2\alpha x + \beta \\ & \frac{1}{v} = 2\alpha x + \beta \\ \end{aligned}$$</p> <p>Next, we differentiate again with respect to time to find the acceleration:</p> <p>$$\begin{aligned} & -\frac{1}{v^2} \frac{dv}{dt} = 2\alpha \frac{dx}{dt} \\ & \text{Since} \quad \frac{dx}{dt} = v, \quad \text{we have:} \\ & -\frac{1}{v^2} \frac{dv}{dt} = 2\alpha v \\ & \frac{dv}{dt} = -2\alpha v^3 \\ & \text{Therefore,} \quad a = -2\alpha v^3 \end{aligned}$$</p>