On an open ground, a motorist follows a track that turns to his left by an angle of $60^{°}$ after every $500\; m$. Starting from a given turn, specify the displacement of the motorist at the third, sixth and eighth turn. Compare the magnitude of the displacement with the total path length covered by the motorist in each case.
On an open ground, a motorist follows a track that turns to his left by an angle of $60^{°}$ after every $500\; m$. Starting from a given turn, specify the displacement of the motorist at the third, sixth and eighth turn. Compare the magnitude of the displacement with the total path length covered by the motorist in each case.
The path followed by the motorist is a regular hexagon with side $500\, m$, as shown in the given figure
Let the motorist start from point $P$. The motorist takes the third turn at $S$.
$\therefore$ Magnitude of displacement $= PS = PV + VS =500+500=1000 \,m$
Total path length $= PQ + QR + RS =500+500+500=1500\, m$
The motorist takes the sixth turn at point $P$, which is the starting point.
$\therefore$ Magnitude of displacement $=0$ Total path length $= PQ + QR + RS + ST + TU + UP$
$=500+500+500+500+500+500=3000 \,m$
The motorist takes the eight turn at point $R$
$\therefore$ Magnitude of displacement $= PR$
$=\sqrt{ PQ ^{2}+ QR ^{2}+2( PQ ) \cdot( QR ) \cos 60^{\circ}}$
$=\sqrt{500^{2}+500^{2}+\left(2 \times 500 \times 500 \times \cos 60^{\circ}\right)}$
$=\sqrt{250000+250000+\left(500000 \times \frac{1}{2}\right)}$
$=866.03\, m$
$\beta=\tan ^{-1}\left(\frac{500 \sin 60^{\circ}}{500+500 \cos 60^{\circ}}\right)=30^{\circ}$
Therefore, the magnitude of displacement is $866.03\, m$ at an angle of $30^{\circ}$ with $PR$. Total path length $=$ Circumference of the hexagon $+ PQ + QR$ $=6 \times 500+500+500=4000\, m$
The magnitude of displacement and the total path length corresponding to the required turns is shown in the given table
| Turn | Magnitude of displacement | Total path length |
| Third | $1000 $ | $1500 $ |
| Sixth | $0 $ | $3000 $ |
| Eighth | $866.03 ; 30^{\circ}$ | $4000$ |
Similar Questions
If $\vec{P}+\vec{Q}=\vec{P}-\vec{Q}$, then
If $\vec{P}+\vec{Q}=\vec{P}-\vec{Q}$, then
Given below in Column $-I$ are the relations between vectors $\vec a \,$ $\vec b \,$ and $\vec c \,$ and in Column $-II$ are the orientations of $\vec a$, $\vec b$ and $\vec c$ in the $XY-$ plane. Match the relation in Column $-I$ to correct orientations in Column $-II$.
Column $-I$
Column $-II$
$(a)$ $\vec a \, + \,\,\vec b \, = \,\,\vec c $
$(i)$ Image
$(b)$ $\vec a \, - \,\,\vec c \, = \,\,\vec b$
$(ii)$ Image
$(c)$ $\vec b \, - \,\,\vec a \, = \,\,\vec c $
$(iii)$ Image
$(d)$ $\vec a \, + \,\,\vec b \, + \,\,\vec c =0$
$(iv)$ Image
Given below in Column $-I$ are the relations between vectors $\vec a \,$ $\vec b \,$ and $\vec c \,$ and in Column $-II$ are the orientations of $\vec a$, $\vec b$ and $\vec c$ in the $XY-$ plane. Match the relation in Column $-I$ to correct orientations in Column $-II$.
| Column $-I$ | Column $-II$ |
| $(a)$ $\vec a \, + \,\,\vec b \, = \,\,\vec c $ | $(i)$ Image |
| $(b)$ $\vec a \, - \,\,\vec c \, = \,\,\vec b$ | $(ii)$ Image |
| $(c)$ $\vec b \, - \,\,\vec a \, = \,\,\vec c $ | $(iii)$ Image |
| $(d)$ $\vec a \, + \,\,\vec b \, + \,\,\vec c =0$ | $(iv)$ Image |
The magnitude of vectors $\overrightarrow{ OA }, \overrightarrow{ OB }$ and $\overrightarrow{ OC }$ in the given figure are equal. The direction of $\overrightarrow{ OA }+\overrightarrow{ OB }-\overrightarrow{ OC }$ with $x$-axis will be
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Match List$- I$ with List$- II.$
$[Image]$
Choose the correct answer from the options given below :
Match List$- I$ with List$- II.$
$[Image]$
Choose the correct answer from the options given below :
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Two vectors $\overrightarrow{ A }$ and $\overrightarrow{ B }$ have equal magnitudes. If magnitude of $\overrightarrow{ A }+\overrightarrow{ B }$ is equal to two times the magnitude of $\overrightarrow{ A }-\overrightarrow{ B }$, then the angle between $\overrightarrow{ A }$ and $\overrightarrow{ B }$ will be .......................
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