Answer

Verified

412.5k+ 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.

Recently Updated Pages

How many sigma and pi bonds are present in HCequiv class 11 chemistry CBSE

Why Are Noble Gases NonReactive class 11 chemistry CBSE

Let X and Y be the sets of all positive divisors of class 11 maths CBSE

Let x and y be 2 real numbers which satisfy the equations class 11 maths CBSE

Let x 4log 2sqrt 9k 1 + 7 and y dfrac132log 2sqrt5 class 11 maths CBSE

Let x22ax+b20 and x22bx+a20 be two equations Then the class 11 maths CBSE

Trending doubts

Which are the Top 10 Largest Countries of the World?

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Write a letter to the principal requesting him to grant class 10 english CBSE

Difference Between Plant Cell and Animal Cell

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

One cusec is equal to how many liters class 8 maths CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

How many crores make 10 million class 7 maths CBSE

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