# How far must you swim upstream to get directly in front of a pizza stand .2 km away on the other...

## Question:

How far must you swim upstream to get directly in front of a pizza stand .2 km away on the other side of a river if you can swim 4 km/hr and the river moves downstream at 1.5 km/hr?

Vector addition consists of the process of adding 2 vectors together, to obtain one resultant vector. Vectors cannot be added straightforward, but have to follow some steps in order to add the adequately. Vectors can be added when they're along the same axis of reference, and as such we have to decompose the vectors to add them. Once the vector have been decomposed into its vertical and horizontal components, then we add up the vectors along the same plane of reference. Finally we use Pythagorean theorem to deduce the resultant vector of the sums of each component.

If the river moves downstream at 1.5 km/h, then the swimmer must move upstream at 1.5 km/h in order to end in front of the pizza stand. We first determine the time it will take the swimmer to reach the other end. We proceed in determining the horizontal component of the speed.

{eq}v^2=v_x^2+v_y^2\\ v_x=\sqrt{v^2-v_y^2}\\ v_x=\sqrt{4^2-1.5^2}\\ v_x=3.71\ m/s\\ {/eq}

Solving for the time:

{eq}v=\dfrac{d}{t}\\ t=\dfrac{d}{v}\\ t=\dfrac{2\ km}{3.71\ km/h}\\ t=0.54\ h {/eq}

Once we have the time, we determine how much the swimmer went upstream.

{eq}d_y=v_y\times t\\ d_y=1.5\ km/h\times 0.54\ h\\ d_y=0.81\ km {/eq} 