A particle is projected from point G such that it touches the points B, C, D and E of a regular hexagon of side ‘a’. Its horizontal range GH is
A. $\sqrt 3 a$
B. $\sqrt 5 a$
C. $\sqrt {7a} $
D. None

Answer
345.6k+ views
Hint: Find the coordinates of the points B, C, D and E and put it in the equation of the parabolic path of the projectile described by the particle. The coordinates of the points B, C, D and E satisfy the equation of the parabola.
Complete step by step answer:Let origin be the midpoint of AF. Then the coordinates of B, C, D and E is given by$B\left( { - a,a\dfrac{{\sqrt 3 }}{2}} \right)$ , $C\left( {\dfrac{{ - a}}{2},a\sqrt 3 } \right)$ , $D\left( {\dfrac{a}{2},a\sqrt 3 } \right)$ , $E\left( {a,a\dfrac{{\sqrt 3 }}{2}} \right)$
The projectile will describe a parabola which is symmetrical about the y-axis. Lets the roots of the path traced by the parabola be r and –r. Then the equation of the parabola is given by $y = k(x - r)(x + r)$ \[ \Rightarrow y = k({x^2} - {r^2})\]
Since the points B and C lie on this parabola, so its coordinates must satisfy the equation of the parabola. Putting the corresponding values of x and y of B in\[y = k({x^2} - {r^2})\], we get
\[a\dfrac{{\sqrt 3 }}{2} = k({a^2} - {r^2})\] ………….(1)
Now, putting corresponding values of x and y of C in\[y = k({x^2} - {r^2})\], we get
\[a\sqrt 3 = k(\dfrac{{{a^2}}}{4} - {r^2})\] ………….(2)
Dividing equations (1) and (2), we get
\[\dfrac{1}{2} = 4\left( {\dfrac{{{a^2} - {r^2}}}{{{a^2} - 4{r^2}}}} \right)\] \[ \Rightarrow {a^2} - 4{r^2} = 8{a^2} - 8{r^2}\]
\[ \Rightarrow 7{a^2} = 4{r^2}\] \[ \Rightarrow r = \dfrac{{a\sqrt 7 }}{2}\]
As the horizontal range is from –r to r, that is 2r.
So, the horizontal range is equal to \[2 \times a\sqrt 7 = \sqrt 7 a\]
Hence, the correct option is (C).
Note:Projectile is the name given to a body thrown with some initial velocity with the horizontal direction and then allowed to move in two dimensions under the influence of gravity. The path followed by a projectile is called its trajectory. The path of a projectile projected horizontally from a point on the ground is a parabola which is symmetrical about the y-axis.
Complete step by step answer:Let origin be the midpoint of AF. Then the coordinates of B, C, D and E is given by$B\left( { - a,a\dfrac{{\sqrt 3 }}{2}} \right)$ , $C\left( {\dfrac{{ - a}}{2},a\sqrt 3 } \right)$ , $D\left( {\dfrac{a}{2},a\sqrt 3 } \right)$ , $E\left( {a,a\dfrac{{\sqrt 3 }}{2}} \right)$
The projectile will describe a parabola which is symmetrical about the y-axis. Lets the roots of the path traced by the parabola be r and –r. Then the equation of the parabola is given by $y = k(x - r)(x + r)$ \[ \Rightarrow y = k({x^2} - {r^2})\]
Since the points B and C lie on this parabola, so its coordinates must satisfy the equation of the parabola. Putting the corresponding values of x and y of B in\[y = k({x^2} - {r^2})\], we get
\[a\dfrac{{\sqrt 3 }}{2} = k({a^2} - {r^2})\] ………….(1)
Now, putting corresponding values of x and y of C in\[y = k({x^2} - {r^2})\], we get
\[a\sqrt 3 = k(\dfrac{{{a^2}}}{4} - {r^2})\] ………….(2)
Dividing equations (1) and (2), we get
\[\dfrac{1}{2} = 4\left( {\dfrac{{{a^2} - {r^2}}}{{{a^2} - 4{r^2}}}} \right)\] \[ \Rightarrow {a^2} - 4{r^2} = 8{a^2} - 8{r^2}\]
\[ \Rightarrow 7{a^2} = 4{r^2}\] \[ \Rightarrow r = \dfrac{{a\sqrt 7 }}{2}\]
As the horizontal range is from –r to r, that is 2r.
So, the horizontal range is equal to \[2 \times a\sqrt 7 = \sqrt 7 a\]
Hence, the correct option is (C).
Note:Projectile is the name given to a body thrown with some initial velocity with the horizontal direction and then allowed to move in two dimensions under the influence of gravity. The path followed by a projectile is called its trajectory. The path of a projectile projected horizontally from a point on the ground is a parabola which is symmetrical about the y-axis.
Recently Updated Pages
Basicity of sulphurous acid and sulphuric acid are

Define absolute refractive index of a medium

Which of the following would not be a valid reason class 11 biology CBSE

Why should electric field lines never cross each other class 12 physics CBSE

An electrostatic field line is a continuous curve That class 12 physics CBSE

What is meant by monosporic development of female class 11 biology CBSE

Trending doubts
Which one of the following places is unlikely to be class 8 physics CBSE

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

What is 1 divided by 0 class 8 maths CBSE

Change the following sentences into negative and interrogative class 10 english CBSE

What is pollution? How many types of pollution? Define it

Difference Between Plant Cell and Animal Cell

Find the HCF and LCM of 6 72 and 120 using the prime class 6 maths CBSE

Give 10 examples for herbs , shrubs , climbers , creepers
