Tanishq crosses the river!

Let the velocity of the river at an instant be $v_{r}$

$\begin{cases} {v_{r}= \left(\dfrac{y}{25}\right), ~~~~~~~~~~~~~~~~~~~~~~y \leq 250\text{m}} \\ {v_{r}= 20- \left(\dfrac{y}{25}\right), ~~~~~~~~~~~~~y>250\text{m}} \end{cases}$

We know that

$\begin{aligned} & y=(5)t &&&& \small \left( \color{#3D99F6} \text{Using } y = vt,\text{ where }v =5 \text{m}s^{-1} \right) &&& \cdots(1) \\ &T=\dfrac{500}{5}=100 \text{s} &&&& \small\left( \color{#3D99F6} \text{T: Total time taken to cross the river} \right) \end{aligned}$

Therefore, in $50$ seconds, Tanishq would cross half the river as his velocity (in the direction perpendicular to the flow of river, which is the only factor that is responsible for him crossing the river) is uniform. Hence we need to calculate the distance(drift) covered in the two halves of the river separately because the velocity of the river is a piece-wise defined function.

$\begin{aligned} &\dfrac{\mathrm{d}x}{\mathrm{d}t}=v_{r} \Rightarrow \displaystyle{\int_{0}^{x}{\mathrm{d}x}=\int_{0}^{50}{\dfrac{y}{25}.\mathrm{d}t}+\int_{50}^{100}{\dfrac{y}{25}.\mathrm{d}t}} && \small \left(\color{#3D99F6}\text{Drift is produced by the velocity of the river current only}\right) \\ & x=\int_{0}^{50}{\dfrac{5t}{25}.\mathrm{d}t}+\int_{50}^{100}{\left(20-\dfrac{5t}{25}\right).\mathrm{d}t} && \small \left( \color{#3D99F6}\text{ Using } (1) \right) \\ & x=\left[\dfrac{t^2}{10} \right]_{0}^{50}+ \left[20t-\dfrac{t^2}{10} \right]_{50}^{100}=250+1000-750=\boxed{500} \end{aligned}$

Note: $x$ is the drift of the swimmer, Tanishq.

First, consider that his velocity increases at a linear rate of $\frac{y}{25}$ , therefore the acceleration is constant up until 250m has been traveled. The next 250m is then a reflection of the first 250m, so the total displacement is twice the displacement at 250m. For the first 250m: u = 0 , v = 250/25 = 10 , t = 250/5 = 50 , s = $\frac{(v+u)}{2}$ *t $\implies$ s = $\frac{10}{2}$ * 50 = 250 . So, in the first 250m, the drift is 250m $\implies$ the total drift is 500m.