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(a) $15\text{ km/h}$

(b) $18\text{ km/h}$

(c) $3\text{ km/h}$

(d) $12\text{ km/h}$

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Hint:For solving this equation we will first understand how we can add two velocity vectors when they are parallel to each other. Then we can answer this question correctly.

__Complete step by step answer:__

Given:

Speed of the swimmer in still water = 9 km/h.

Speed of flow of the river = 6 km/h.

Now, we have to find the downstream speed of the swimmer. Which means when the swimmer will swim in the direction of flow of the river. Then, the velocity vector of the swimmer and a velocity vector of the flow of the river will be parallel to each other. For more clarity look at the below figure:

In the above figure, there are three velocity vectors represented. On the left-hand side, the smaller velocity vector is the velocity vector of the velocity of flow of the river and the larger velocity vector is the velocity vector of the velocity of the swimmer. On the right-hand side, there is only one velocity vector which represents the resultant of the two velocity vectors on the left-hand side and it is the velocity vector of the downstream velocity of the swimmer.

Now, let $\overrightarrow{{{V}_{S}}}$ is the velocity vector of the swimmer velocity in still water, $\overrightarrow{{{V}_{W}}}$ is the velocity vector of the flow of the river and $\overrightarrow{{{V}_{D}}}$ is the downstream velocity vector. Then,

$\overrightarrow{{{V}_{D}}}=\overrightarrow{{{V}_{S}}}+\overrightarrow{{{V}_{W}}}$

Now, as we know that $\overrightarrow{{{V}_{S}}}$ is parallel to $\overrightarrow{{{V}_{W}}}$ . Then magnitude of $\overrightarrow{{{V}_{D}}}$ = 9 + 6 = 15 km/h.

Thus, the downstream speed of the swimmer will be 15 km/h.

Hence, (a) is the correct option.

Note: Although the question is very easy to solve but the student should not confuse the word downstream with the word upstream. Both are opposite to each other in that the upstream swimmer swims in the opposite direction to the flow of the river and the downstream swimmer swims with the flow in the direction of flow of the river. So, we should add both speeds to get the downstream speed and not to subtract them.

Given:

Speed of the swimmer in still water = 9 km/h.

Speed of flow of the river = 6 km/h.

Now, we have to find the downstream speed of the swimmer. Which means when the swimmer will swim in the direction of flow of the river. Then, the velocity vector of the swimmer and a velocity vector of the flow of the river will be parallel to each other. For more clarity look at the below figure:

In the above figure, there are three velocity vectors represented. On the left-hand side, the smaller velocity vector is the velocity vector of the velocity of flow of the river and the larger velocity vector is the velocity vector of the velocity of the swimmer. On the right-hand side, there is only one velocity vector which represents the resultant of the two velocity vectors on the left-hand side and it is the velocity vector of the downstream velocity of the swimmer.

Now, let $\overrightarrow{{{V}_{S}}}$ is the velocity vector of the swimmer velocity in still water, $\overrightarrow{{{V}_{W}}}$ is the velocity vector of the flow of the river and $\overrightarrow{{{V}_{D}}}$ is the downstream velocity vector. Then,

$\overrightarrow{{{V}_{D}}}=\overrightarrow{{{V}_{S}}}+\overrightarrow{{{V}_{W}}}$

Now, as we know that $\overrightarrow{{{V}_{S}}}$ is parallel to $\overrightarrow{{{V}_{W}}}$ . Then magnitude of $\overrightarrow{{{V}_{D}}}$ = 9 + 6 = 15 km/h.

Thus, the downstream speed of the swimmer will be 15 km/h.

Hence, (a) is the correct option.

Note: Although the question is very easy to solve but the student should not confuse the word downstream with the word upstream. Both are opposite to each other in that the upstream swimmer swims in the opposite direction to the flow of the river and the downstream swimmer swims with the flow in the direction of flow of the river. So, we should add both speeds to get the downstream speed and not to subtract them.