Answer to Question #95609 in Mechanics | Relativity for Omar Bari

Question #95609
A man runs 500m at 37° North by east then runs another 1000m at 60°west by north. Determine the module of the run and the orientation.
1
Expert's answer
2019-10-01T12:50:19-0400

First a diagram of the situation is made.





The components are obtained for each displacement.



# Displacement 1


Horizontal axis component


cos(370)=D1xD1D1x=D1cos(370)D1x=500mcos(370)D1x=399mcos(37^{0})=\frac{D_{1x}}{D_{1}} \\D_{1x}=D_{1}cos(37^{0}) \\ D_{1x}=500m*cos(37^{0}) \\D_{1x}=399m


Vertical axis component

sin(370)=D1yD1D1y=D1sin(370)D1y=500msin(370)D1y=301msin(37^{0})=\frac{D_{1y}}{D_{1}} \\D_{1y}=D_{1}sin(37^{0}) \\ D_{1y}=500m*sin(37^{0}) \\D_{1y}=301m


Expressing as a vector


D1=(399i+301j)[m]\vec{D_{1}}=(399i+301j)[m]


# Displacement 2


Horizontal axis component


sin(600)=D2xD2D2x=D2sin(600)D2x=1000msin(600)D2x=866msin(60^{0})=\frac{D_{2x}}{D_{2}} \\D_{2x}=D_{2}sin(60^{0}) \\ D_{2x}=1000m*sin(60^{0}) \\D_{2x}=866m


Vertical axis component


cos(600)=D2yD2D2y=D2cos(600)D2y=1000mcos(600)D2y=500mcos(60^{0})=\frac{D_{2y}}{D_{2}} \\D_{2y}=D_{2}cos(60^{0}) \\ D_{2y}=1000m*cos(60^{0}) \\D_{2y}=500m


Expressing as a vector


D2=(866i+500j)[m]\vec{D_{2}}=(-866i+500j)[m]


The total displacement is


\vec{} DR=D1+D2DR=(399i+301j)[m]+(866i+500j)[m]DR=[(399866)i+(301+500)j][m]DR=(467i+801)[m]\vec{D_{R}}=\vec{D_{1}}+\vec{D_{2}} \\ \vec{D_{R}}=(399i+301j)[m] +(-866i+500j)[m]\\ \vec{D_{R}}=[(399-866)i+(301+500)j][m] \\ \vec{D_{R}}=(-467i+801)[m]


The run module is calculated using the Pythagorean theorem.


DR=(467)2+(801)2DR=927m|\vec{D_{R}}|=\sqrt{(-467)^{2}+(801)^{2}} \\ |\vec{D_{R}}|=927m


the module of the run is DR=927m\boxed{|\vec{D_{R}}|=927m}


The direction with respect to the + i axis is


tanθ=(801467)θ=tan1(801467)θ=59.750+1800=1200tan\theta=(\frac{801}{-467}) \\ \theta=tan^{-1}(\frac{801}{-467}) \\ \theta=-59.75^{0}+180^{0}=120^{0} Measured from the East direction (+ i)


Note: Being a negative component, the resulting angle will be reflected as in quadranta IV, but 180 degrees must be added to reflect it in quadrant II


with respect to the north the angle is 1200900=300120^{0}-90^{0}=30^{0}


Expressing the closest coordinate axis.


30°west by north


the the orientation of the run is  30°west by north\boxed{ \text{ 30°west by north}}


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