A fluid flows along the \(x\) axis with a velscity given by \(\mathbf{V}=(x / t) \hat{\mathbf{i}},\) where \(x\) is in feet and \(t\) in seconds. (a) Plot the speed for \(0 \leq x \leq 10\) ft and \(t=3\) s. (b) Plot the speed for \(x=7\) ft and \(2 \leq t \leq 4 \mathrm{s},\) (c) Determine the local and convective acceleration. (d) Show that the acceleration of any fluid particle in the flow is zero. (e) Explain physically how the velocity of a particle in this unsteady flow remains constant throughout its motion.

For part (a) and (b), plot speed for \(0 \leq x \leq 10\) ft when \(t=3\) s and for \(x=7\) ft when \(2 \leq t \leq 4\) s. Use the given function \(\mathbf{V}=\frac{x}{t}\) to calculate these values and then plot.

For part (c), the acceleration of a fluid particle may be divided into local and convective acceleration. Local acceleration is the rate of change of velocity at a point as time advances. It can be calculated by taking the derivative of velocity with respect to time: \(\frac{{dv}}{{dt}}\) whereas convective acceleration is the acceleration of a fluid particle in the flow direction. It involves change in speed and direction as the particle moves from one point to another It can be calculated by taking the product of velocity and derivative of velocity with respect to space: \(u\frac{{du}}{{dx}}\). Here, since velocity function \(\mathbf{V}=\frac{x}{t}\), let's derivate it against time \(t\) for local acceleration and against space \(x\) for convective acceleration.

Now, for part (d), the total acceleration of a particle is the sum of local and convective accelerations. Since we have found the local and convective accelerations in previous step, sum up both accelerations. The aim is to show that the total acceleration of any fluid particle in the flow is zero.

Finally, for part (e), provide a physical explanation for why the velocity of a particle in this unsteady flow remains constant throughout its motion, based on the results in previous steps. The key element here is to understand how the velocity changes with respect to time and space are balanced out in such a way that the total velocity remains constant.