To find the average speed of the vehicle, we need to consider the total distance traveled and the total time taken.$\\$Let's denote:$\\$• $d$ as the total distance traveled by the vehicle.$\\$• $v$ as the speed for the first half of the distance.$\\$• $2v$ as the speed for the second half of the distance.$\\$The vehicle travels half the distance $\left(\frac{d}{2}\right)$ with speed $v. \\$Time taken for the first half $t_1$ is given by:$\\ t_1 = \frac{\frac{d}{2}}{v} = \frac{d}{2v} \\$The vehicle travels the remaining half $\left(\frac{d}{2}\right)$ with speed $2v. \\$Time taken for the second half $t_2$ is given by:$\\ t_2 = \frac{\frac{d}{2}}{2v} = \frac{d}{4v} \\$The total time $t_{total}$ is the sum of $t_1$ and $t_2: \\ t_{total} = t_1 + t_2 = \frac{d}{2v} + \frac{d}{4v} \\$Combine the terms:$\\ t_{total} = \frac{2d}{4v} + \frac{d}{4v} = \frac{3d}{4v} \\$The average speed $v_{avg}$ is given by the total distance divided by the total time:$\\ v_{avg} = \frac{d}{t_{total}} = \frac{d}{\frac{3d}{4v}} \\$Simplify the expression:$\\ v_{avg} = \frac{d \cdot 4v}{3d} = \frac{4v}{3} \\$Therefore, the average speed of the vehicle is $\frac{4v}{3}. \\$This corresponds to Option 3.