Question

the three displacements given are 69.8m , 53? west of north; 11.3m straight north; and 83.6m , 42? south of east. What magnitude does the winner calculate for the displacement to find the keys to the Porsche?

Answer 1

Three displacements vectors which are given as :

A = 69.8 m                      (_A = 53⁰ west of north)

B = 83.6 m                      (_B = 42⁰ south of east)

C = 11.3 m                      (_C = straight north)

The resultant displacement is given as :

= + +                     { eq. 1 }

The angles of the vectors, measured from the x-axis towards the y-axis are given as :

_A = (90 - 53⁰) = 37 degree

_B = (180 + 42⁰) = 222 degree

_C = 270 degree

the components of on x & y-axis which is given as :

A_x = A Cos                    { eq. 2 }

inserting the values in eq.2

A_x = (69.8 m) Cos (37⁰)

A_x = 55.74 m

A_y = A Sin                         { eq. 3 }

inserting the values in eq.3

A_y = (69.8 m) Sin (37⁰)

A_y = 42 m

the components of on x & y-axis which is given as :

B_x = B Cos       

B_x = (83.6 m) Cos (222⁰)

B_x = - 62.12 m

and B_y = B Sin        

B_y = (83.6 m) Sin (222⁰)

B_y = - 55.9 m

the components of on x & y-axis which is given as :

C_x = C Cos       

C_x = (11.3 m) Cos (270⁰)

C_x = 0

and C_y = C Sin

C_y = (11.3 m) Sin (270⁰)

C_y = - 11.3 m

on the x-axis, resultant value is given as :

R_x = A_x + B_x + C_x                           { eq. 4 }

inserting the values in eq.4,

R_x = (55.74 m) + (-62.12 m) + (0 m)

R_x = - 6.38 m

on the y-axis, resultant value is given as :

R_y = A_y + B_y + C_y { eq. 5 }

inserting the values in eq.5,     

R_y = (42 m) + (- 55.9 m) + (-11.3 m)

R_y = - 25.2 m

the magnitude of the net resultant displacement is given as :

R = R_x² + R_y²                         { eq.6 }

inserting the values in eq.6

R = (-6.38 m)² + (-25.2 m)²

R = 675.74 m

R = 26 m

and the direction is given as :

= tan^-1 (R_y / R_x) { eq.7 }

inserting the values in eq.7

= tan^-1 [(- 25.2 m) / (- 6.38 m)]

= tan^-1 (3.94)

= 75.7 degree