A tourist follows a passage that takes her 160 m west, then 180.m at an angle of 45.0∘ south of east, and finally 250 m at an angle 35.0∘ north of east. The total journey takes 12 minutes. (4) a. Calculate the magnitude of her displacement from her original position.

b. She measures the distance she has walked to a precision of 5%. She times her total journey to ±20 s. (4)

(i) What is her average speed?

(ii) What is the absolute uncertainty on her absolute speed?

a) R = 172.82 m, 5.3 north of east    b)   v = (0.24 ± 0.01) m / s

Explanation:

a) the displacement is a vector, so the easiest method to lock is looking for each component

let's start decomposing the vectors

 x₁ = - 160 m

second shift

angle 45 south of east

     cos (-45) = x₂ / d₂

     sin (-45) = y₂ / d₂

     x₂ = d₂ cos 45

     y₂ = -d₂ sin 45

     x₂ = 180 cos 45 = 127.28 m

     y₂ = -180 sin 45 = - 127.28 m

third shift

      cos 35 = x₃ / d₃

      sin 35 = y₃ / d₃

      x₃ = d₃ cos 35

      y₃ = d₃ sin 35

      x₃ = 250 cos 35 = 204.79 m

      y₃ = 250 sin 35 = 143.39 m ₃

X axis

       x_total = x₁ + x₂ + x₃

       x_total = -160 +127.28 +204.79

       x_total = 172.07 m

Y axis  

      y_total = y₁ + y₂ + y₃

      y_total = 0 - 127.28 + 143.39

      y_total = 16.11 m

to compose the displacement we use the Pythagorean theorem

      R =

      R =

      R = 172.82 m

in angle is

      tan θ = y_total / x_total

      ten θ = 16.11 / 172.07 = 0.0936

      θ = tan⁻¹ 0.0936

       θ = 5.3

angle is 5.3 north of east

b) the error in the distance is 5%,

               e% = ΔR/R  100

               ΔR = e% R / 100

               ΔR = 5 172.82 / 100

               ΔR = 8.6 m

the time error Dt = 20 s

We calculate the speed and this we calculate the error

      v = R / t

      v = 172.82 / 12 60

      v = 0.240027 m / s

the error in this magnitude is

      Δv =

       Δv =

         Δv =

         Δv = 0.013 m / s

the measurement result is

         v = (0.24 ± 0.01) m / s

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