Answer:
Magnitude of avg velocity, ![|v_{avg}| = 18.9 km/h](https://tex.z-dn.net/?f=%7Cv_%7Bavg%7D%7C%20%3D%2018.9%20km%2Fh)
![\theta' = 56.85^{\circ}](https://tex.z-dn.net/?f=%5Ctheta%27%20%3D%2056.85%5E%7B%5Ccirc%7D)
Given:
Constant speed of train, v = 79 km/h
Time taken in East direction, t = 27 min = ![\frac{27}{60} h](https://tex.z-dn.net/?f=%5Cfrac%7B27%7D%7B60%7D%20h)
Angle, ![\theta = 50^{\circ}](https://tex.z-dn.net/?f=%5Ctheta%20%3D%2050%5E%7B%5Ccirc%7D)
Time taken in
east of due North direction, t' = 29 min = ![\frac{29}{60} h](https://tex.z-dn.net/?f=%5Cfrac%7B29%7D%7B60%7D%20h)
Time taken in west direction, t'' = 37 min = ![\frac{27}{60} h](https://tex.z-dn.net/?f=%5Cfrac%7B27%7D%7B60%7D%20h)
Solution:
Now, the displacement, 's' in east direction is given by:
![\vec{s} = vt = 79\times \frac{27}{60} = 35.5\hat{i} km](https://tex.z-dn.net/?f=%5Cvec%7Bs%7D%20%3D%20vt%20%3D%2079%5Ctimes%20%5Cfrac%7B27%7D%7B60%7D%20%3D%2035.5%5Chat%7Bi%7D%20km)
Displacement in
east of due North for 29.0 min is given by:
![\vec{s'} = vt'sin50^{\circ}\hat{i} + vt'cos50^{\circ}\hat{j}](https://tex.z-dn.net/?f=%5Cvec%7Bs%27%7D%20%3D%20vt%27sin50%5E%7B%5Ccirc%7D%5Chat%7Bi%7D%20%2B%20vt%27cos50%5E%7B%5Ccirc%7D%5Chat%7Bj%7D)
![\vec{s'} = 79(\frac{29}{60})sin50^{\circ}\hat{i} + 79(\frac{29}{60})cos50^{\circ}\hat{j}](https://tex.z-dn.net/?f=%5Cvec%7Bs%27%7D%20%3D%2079%28%5Cfrac%7B29%7D%7B60%7D%29sin50%5E%7B%5Ccirc%7D%5Chat%7Bi%7D%20%2B%2079%28%5Cfrac%7B29%7D%7B60%7D%29cos50%5E%7B%5Ccirc%7D%5Chat%7Bj%7D)
![\vec{s'} = 29.25\hat{i} + 24.54\hat{j} km](https://tex.z-dn.net/?f=%5Cvec%7Bs%27%7D%20%3D%2029.25%5Chat%7Bi%7D%20%2B%2024.54%5Chat%7Bj%7D%20km)
Now, displacement in the west direction for 37 min:
![\vec{s''} = - vt''hat{i} = - 79\frac{37}{60} = - 48.72\hat{i} km](https://tex.z-dn.net/?f=%5Cvec%7Bs%27%27%7D%20%3D%20-%20vt%27%27hat%7Bi%7D%20%3D%20-%2079%5Cfrac%7B37%7D%7B60%7D%20%3D%20-%2048.72%5Chat%7Bi%7D%20km)
Now, the overall displacement,
![\vec{s_{net}} = \vec{s} + \vec{s'} + \vec{s''}](https://tex.z-dn.net/?f=%5Cvec%7Bs_%7Bnet%7D%7D%20%3D%20%5Cvec%7Bs%7D%20%2B%20%5Cvec%7Bs%27%7D%20%2B%20%5Cvec%7Bs%27%27%7D)
![\vec{s_{net}} = 35.5\hat{i} + 29.25\hat{i} + 24.54\hat{j} - 48.72\hat{i}](https://tex.z-dn.net/?f=%5Cvec%7Bs_%7Bnet%7D%7D%20%3D%2035.5%5Chat%7Bi%7D%20%2B%2029.25%5Chat%7Bi%7D%20%2B%2024.54%5Chat%7Bj%7D%20-%2048.72%5Chat%7Bi%7D)
![\vec{s_{net}} = 16.03\hat{i} + 24.54\hat{j} km](https://tex.z-dn.net/?f=%5Cvec%7Bs_%7Bnet%7D%7D%20%3D%20%2016.03%5Chat%7Bi%7D%20%2B%2024.54%5Chat%7Bj%7D%20km)
(a) Now, average velocity,
is given:
![v_{avg} = \frac{total displacement, \vec{s_{net}}}{total time, t}](https://tex.z-dn.net/?f=v_%7Bavg%7D%20%3D%20%5Cfrac%7Btotal%20displacement%2C%20%5Cvec%7Bs_%7Bnet%7D%7D%7D%7Btotal%20time%2C%20t%7D)
![v_{avg} = \frac{16.03\hat{i} + 24.54\hat{j}}{\frac{27 + 29 + 37}{60}}](https://tex.z-dn.net/?f=v_%7Bavg%7D%20%3D%20%5Cfrac%7B16.03%5Chat%7Bi%7D%20%2B%2024.54%5Chat%7Bj%7D%7D%7B%5Cfrac%7B27%20%2B%2029%20%2B%2037%7D%7B60%7D%7D)
![v_{avg} = 10.34\hat{i} + 15.83\hat{j}) km/h](https://tex.z-dn.net/?f=v_%7Bavg%7D%20%3D%2010.34%5Chat%7Bi%7D%20%2B%2015.83%5Chat%7Bj%7D%29%20km%2Fh)
Magnitude of avg velocity is given by:
![|v_{avg}| = \sqrt{(10.34)^{2} + (15.83)^{2}} = 18.9 km/h](https://tex.z-dn.net/?f=%7Cv_%7Bavg%7D%7C%20%3D%20%5Csqrt%7B%2810.34%29%5E%7B2%7D%20%2B%20%2815.83%29%5E%7B2%7D%7D%20%3D%2018.9%20km%2Fh)
(b) angle can be calculated as:
![tan\theta' = \frac{15.83}{10.34}](https://tex.z-dn.net/?f=tan%5Ctheta%27%20%3D%20%5Cfrac%7B15.83%7D%7B10.34%7D)
![\theta' = tan^{- 1}\frac{15.83}{10.34} = 56.85^{\circ}](https://tex.z-dn.net/?f=%5Ctheta%27%20%3D%20tan%5E%7B-%201%7D%5Cfrac%7B15.83%7D%7B10.34%7D%20%3D%2056.85%5E%7B%5Ccirc%7D)