Question

While surveying a cave, a spelunker follows a passage 180 m straight west, then 220 m in a direction 45.0 ∘ east of south, and then 260 m at 30.0 ∘ east of north. After a fourth unmeasured displacement, she finds herself back where she started. Use vector components to find the magnitude and direction of the fourth displacement.

Answer 1

let the spelunker starting be 0 i + 0 j here i and j are unit vectors is +x and +y direction.

Now when the spelunker travels west 180 meters hence his co ordinates becomes -180 i + 0 j.

Now spelunker travels 45 degree east of south i.e. he makes 45 degree with + x axis downwards. Hence his new cordinates are

(-180+220 cos45) i + (0- 220 sin45) j.

= -24 i -156 j .

Now the spelunker travels 260m making 30 degree with east of north i.e. making 60 degree with +x axis upwards. hence his new cordinates are

=(-24+ 260 cos60 ) i + (-156 + 260 sin60 )j . = 106 i + 65 j .

hence the distance the final vector has to travel to reach back to same position = 106 i + 65 j and slope of last vector

=tan^-1 (106/65) = 58 degree with + axis.

Magnitude of final vector = = 124.34m