Question

8. An ant on a picnic table travels 30 cm eastward, then 25 cm northward, and finally 15 cm westward. What is the ant's displacement, angle, and direction
relative to its original position?

Answer 1

Answer:

The displacement of the ant, R = 29.15 cm

The angle of the resultant displacement with its original position is, θ = 30° 57'

The direction of the displacement is towards the northeast.  

Explanation:

Given data,

The displacement towards east, d₁ = 30 cm

The displacement towards north, d₂ = 25 cm

The displacement towards  west, d₃ = 15 cm

The total displacement towards east,

                                       d₄ = d₁ - d₃

                                           = 30 - 15

                                           = 15 cm

The total displacement of ant is given by the resultant displacement,

                            R = √(d₂² + d₄² + 2· d₂ d₄ CosФ)

Where Ф is the angle between the vectors, d₂ & d₄

                                                Ф = 90°

Therefore,

                               R = √(d₂² + d₄²)

Substituting in the above equation,

                               R = √(25² + 15²)

                                  = 29.15 cm

Hence, the displacement of the ant, R = 29.15 cm

The angle of the resultant displacement with its original position is,

                                θ = tan⁻¹ (d₄ / d₂)

                                   = tan⁻¹ (15 / 25)

                                   = tan⁻¹ 0.6

                                   = 30° 57'

Hence, the angle of the resultant displacement the its original position is, θ = 30° 57'

The direction of the displacement is towards the northeast.