Animated Solution for Physics - Kinematics: A person is swimming with a speed of $10\text{m/s}$ at an angle of $120^{\circ }$ with the flow and reaches to a point directly opposite on the other side of the river. The speed of the flow is $x\text{m/s}$. The value of $x$ to the nearest integer is ……… .

Imagine you are standing on the bank of a river, staring at a point exactly opposite to you on the other side. Your goal is to reach that exact spot. However, the river isn't still; it's flowing with a certain speed, $v_r=x\text{m/s}$. If you were to just dive in and swim straight across, the river's current would sweep you downstream, and you'd miss your target entirely.

To counteract this, you must swim at an angle against the current. The problem states that you are swimming with a speed of $v_{mr}=10\text{m/s}$ relative to the water, at an angle of $120^{\circ }$ with the direction of the river's flow.

Let's set up a coordinate system. Let the river flow along the positive X-axis. The point you want to reach lies directly along the positive Y-axis. Your actual path across the river is your resultant velocity, ${\vec{v}}_s$. For you to reach the directly opposite point, ${\vec{v}}_s$ must point strictly along the Y-axis.

Your swimming velocity, ${\vec{v}}_{mr}$, makes an angle of $\alpha =120^{\circ }$ with the X-axis. We can break this angle down. The angle between the X-axis and the Y-axis is $90^{\circ }$. Therefore, the angle your swimming velocity makes with the Y-axis (the vertical) is:

Here is the beautiful physics behind the problem: your swimming velocity ${\vec{v}}_{mr}$ has two components. The vertical component ($v_{mr}\cos \theta$) is what actually takes you across the river. The horizontal component ($v_{mr}\sin \theta$) is fighting against the river's flow.

For you to travel strictly along the Y-axis, your horizontal velocity must be exactly zero. This means the horizontal component of your swimming velocity must perfectly cancel out the river's flow velocity:

Alternatively, if you look at the vector triangle formed by ${\vec{v}}_{mr}$, ${\vec{v}}_r$, and ${\vec{v}}_s$, you can see that ${\vec{v}}_r$ is the side opposite to the angle $\theta$, and ${\vec{v}}_{mr}$ is the hypotenuse. Thus, using basic trigonometry:

Now, we simply substitute the values we know into our master equation. We know $\theta =30^{\circ }$, $v_{mr}=10\text{m/s}$, and $v_r=x$.

We know from trigonometry that $\sin (30^{\circ })=\frac{1}{2}$. Plugging this in:

Multiplying both sides by 10, we find the speed of the river:

And there we have it! The river is flowing at exactly $5\text{m/s}$.

Food for thought: Notice that this is only possible because your swimming speed ($10\text{m/s}$) is greater than the river's speed ($5\text{m/s}$). If the river were flowing faster than you could swim, $\sin \theta$ would be greater than 1, which is mathematically impossible. In that scenario, no matter what angle you choose, you would inevitably be swept downstream!