Physics Practice Problems Relative Motion Practice Problems Solution: A swimmer is capable of swimming 0.80 m/s in still...

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# Solution: A swimmer is capable of swimming 0.80 m/s in still water.(a) At what upstream angle must the swimmer aim, if she is to arrive at a point directly across a 55 m wide river whose current is 0.50 m/s ?(b) How long will it take her?

###### Problem

A swimmer is capable of swimming 0.80 m/s in still water.

(a) At what upstream angle must the swimmer aim, if she is to arrive at a point directly across a 55 m wide river whose current is 0.50 m/s ?

(b) How long will it take her?

###### Solution

Whenever we have a problem about a boat or swimmer crossing a river, it's safe to assume it's a two-dimensional relative motion problem. The steps to solve a problem like this are going to be:

1. Organize information: the variables involved in a problem like this are velocities, distances, and time. It's always a good idea to draw yourself a diagram and label knowns to help you visualize the information!
2. Combine velocities.
3. Solve for the target variable.

In general, the equation we'll use to add velocities is:

$\overline{){\stackrel{\mathbf{⇀}}{\mathbit{v}}}_{\mathbit{P}\mathbit{A}}{\mathbf{=}}{\stackrel{\mathbf{⇀}}{\mathbit{v}}}_{\mathbit{P}\mathbit{B}}{\mathbf{+}}{\stackrel{\mathbf{⇀}}{\mathbit{v}}}_{\mathbit{B}\mathbit{A}}}$

In this problem, the swimmer has some velocity compared to the water of the river, and the river also moves with a velocity relative to the ground. If someone on the riverbank measured the swimmer's velocity, it would be the sum of those two vectors—we'll call that the "effective velocity" or veff.

$\overline{){\stackrel{\mathbf{⇀}}{\mathbit{v}}}_{\mathbit{e}\mathbit{f}\mathbit{f}}{\mathbf{=}}{\stackrel{\mathbf{⇀}}{\mathbit{v}}}_{{\mathbit{s}}}{\mathbf{+}}{\stackrel{\mathbf{⇀}}{\mathbit{v}}}_{{\mathbit{r}}}}$

We may also need the equation relating constant velocity to displacement:

$\overline{)\stackrel{\mathbf{⇀}}{\mathbit{v}}{\mathbf{=}}\frac{\mathbf{∆}\stackrel{\mathbf{⇀}}{\mathbit{r}}}{\mathbf{∆}\mathbit{t}}}$

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