How Does a Fire Ant Calculate Its Displacement?

[AnkhUNC]

[SOLVED] Displacement over a vector

Homework Statement

A fire ant, searching for hot sauce in a picnic area, goes through three displacements along level ground: 1 for 0.46 m southwest (that is, at 45° from directly south and from directly west), 2 for 0.59 m due east, and 3 for 0.52 m at 60° north of east. Let the positive x direction be east and the positive y direction be north. What are (a) the x component and (b) the y component of 1? What are (c) the x component and (d) the y component of 2? What are (e) the x component and (f) the y component of 3? What are (g) the x component and (h) the y component, (i) the magnitude, and (j) the direction of the ant's net displacement? If the ant is to return directly to the starting point, (k) how far and (l) in what direction should it move? Give all angles as positive (counterclockwise) angles relative to the +x-axis.

Homework Equations

The Attempt at a Solution

I have everything solved up until (l)


(g).5247308807m
(h).124642017m
(i)5393311873m
(j)13.36212366° (degrees)
(k).539311873m

But shouldn't (l) be the same angle as (j)? Or am I just missing something.

 

[anathema]

Hello,

Great job on solving the previous parts of the problem! For part (l), you are correct in thinking that the angle should be the same as (j). However, it is important to remember that the angle should be measured from the +x-axis, not the +y-axis. So, the angle for (l) should be the same as (j) but measured from the +x-axis. Keep up the good work!

 

[Moetasim]

I understand that displacement is a vector quantity that describes the change in position of an object. In this scenario, the fire ant is undergoing multiple displacements, each with its own magnitude and direction. The x and y components of each displacement can be determined by using trigonometric functions and the given angles and distances.

(a) The x component of 1 can be found by multiplying the magnitude (0.46 m) by the cosine of 45 degrees, giving us a value of 0.325 m.

(b) Similarly, the y component of 1 can be found by multiplying the magnitude (0.46 m) by the sine of 45 degrees, giving us a value of 0.325 m.

(c) The x component of 2 is simply the magnitude (0.59 m) since it is in the positive x direction.

(d) The y component of 2 is 0 since the displacement is solely in the x direction.

(e) The x component of 3 can be found by multiplying the magnitude (0.52 m) by the cosine of 60 degrees, giving us a value of 0.26 m.

(f) The y component of 3 can be found by multiplying the magnitude (0.52 m) by the sine of 60 degrees, giving us a value of 0.45 m.

(g) To find the x component of the net displacement, we add the x components of each individual displacement (0.325 m + 0.59 m + 0.26 m), giving us a value of 1.175 m.

(h) To find the y component of the net displacement, we add the y components of each individual displacement (0.325 m + 0 + 0.45 m), giving us a value of 0.775 m.

(i) The magnitude of the net displacement can be found using the Pythagorean theorem, c^2 = a^2 + b^2, where c is the magnitude and a and b are the x and y components, respectively. This gives us a value of 1.376 m.

(j) The direction of the net displacement can be found using the inverse tangent function, tan^-1(b/a), where b and a are the y and x components, respectively. This gives us a value of 32.1 degrees, or 32.1 degrees counterclockwise from the +x-axis