Question

Question #338d5

Answer 1

`"magnitude" = 48.6` `"m"`

`"direction" = 64.4^"o"color(white)(l)"south of east"`

Explanation:

We're asked to find the total displacement of the dog, given two constituent displacements it took.

To do this, what we can do is add the two displacement vectors together to get the net displacement. In order to do that, we must first split each displacement into components.

The first displacement has magnitude `24` `"m"` and direction `12^"o"` west of south. We want our angles to be measured anticlockwise from the positive `x`-axis (east), which will be `ul(-102^"o"`:

The components are given by

`x_2 = (24color(white)(l)"m")cos(-102^"o") = color(red)(ul(-4.99color(white)(l)"m"`

`y_1 = (24color(white)(l)"m")sin(-102^"o") = color(green)(ul(-23.5color(white)(l)"m"`

Similarly for the second displacement:

`x_2 = (33color(white)(l)"m")cos(-38^"o") = color(red)(ul(26.0color(white)(l)"m"`

`y_2 = (33color(white)(l)"m")sin(-38^"o") = color(green)(ul(-20.3color(white)(l)"m"`

Now we add the respective componentstogether to find the components of the net displacement:

`x = color(red)(-4.99color(white)(l)"m") + color(red)(26.0color(white)(l)"m") = color(crimson)(ul(21.0color(white)(l)"m"`

`y = color(green)(-23.5color(white)(l)"m") + (color(green)(-20.3color(white)(l)"m")) = color(yellowgreen)(ul(-43.8color(white)(l)"m"`

The magnitude of the displacement is

`r = sqrt(x^2 + y^2) = sqrt((color(crimson)(21.0color(white)(l)"m"))^2 + (color(yellowgreen)(-43.8color(white)(l)"m"))^2) = color(blue)(ulbar(|stackrel(" ")(" "48.6color(white)(l)"m"" ")|)`

and the direction is

`theta = arctan(y/x) = arctan((color(yellowgreen)(-43.8color(white)(l)"m"))/(color(crimson)(21.0color(white)(l)"m"))) = color(blue)(ulbar(|stackrel(" ")(" "64.4^"o"color(white)(l)"south of east"" ")|)`

Always make sure your arctangent calculation is pointing in the correct direction and not `180^"o"` off!