Question

An ant, crazed by the Sun on a hot Texas afternoon, darts over an xy plane scratched in the dirt. The x and y components of four consecutive darts are the following, all in centimeters: (33.0, 48.0), (b_x, -71.0), (-22.0, c_y), (-86.0, -60.0).The overall displacement of the four darts has the xy components (-120, -29.0).What are (a) b_x and (b) c_y? What are the (c) magnitude and (d) angle (counterclockwise from the positive direction of the x axis) of the overall displacement?

Answer 1

Suppose, X-component of the overall displacement = Dx

and Y-component of the overall displacement = Dy

(a) We have -

Dx = ax + bx + cx + dx

=> -120 = 33.0 + bx -22.0 - 86.0

=> -120 = -75.0 + bx

=> bx = -45 cm

(b) Dy = ay + by + cy + dy

=> -29.0 = 48.0 -71.0 + cy - 60.0

=> -29.0 = -83.0 + cy

=> cy = 54.0 cm

(c) Magnitude, D = Sqrt[Dx^2 + Dy^2]

= Sqrt[(-120)^2 + (-29)^2]

= 123.5 cm

(d) Tan = Dy/Dx

= -120 / -29

= 4.14

=> = tan^-1(4.14) = 76.4 deg

Since, for both x and y cordinates are negative. So, this angle lies in the third quadrant.

So, total angle in counterclockwise from positive direction of x-axis = 180 + 76.4 = 256.4 deg. (Answer)