Question

A carrom board (4 ft x 4 ft square) has the queen at the
centre. The queen, hit by the striker moves to the front
edge, rebounds and goes in the hole behind the striking
line. Find the magnitude of displacement of the queen
(a) from the centre to the front edge, (b) from the front
edge to the hole and (c) from the centre to the hole.​

Answer 1

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When a queen hit by striker from the centre it displaced from O to A then it rebounds and goes in the hole at B.Draw a normal at point A, using law of reflection angle i = angle r. Draw a perpendicular from point O at point C . Let AC=X then AD=2-X

(a)    To find magnitude of displacement of queen

       From the figure,

      begin mathsize 14px style 2 over straight X equals fraction numerator 4 over denominator 2 minus straight X end fraction 2 minus straight X equals 2 straight X 3 straight X equals 2 straight X equals 2 over 3 Magnitude space straight s subscript 1 equals 2 over 3 straight i with hat on top plus 2 straight j with hat on top open vertical bar straight s subscript 1 close vertical bar equals square root of 4 over 9 plus 4 end root equals 2 over 3 square root of 10 equals 2.02 space ft end style

(b) From the front edge to the hole

    When the queen rebounds and goes to the hole, its displacement is AC. From the traingle triangleCDA. Hence,

    begin mathsize 14px style stack straight s subscript 2 with rightwards arrow on top equals open parentheses 2 minus 2 over 3 close parentheses straight i with overparenthesis on top minus 4 straight j with overparenthesis on top space space space space equals 4 over 3 straight i with overparenthesis on top minus 4 straight j with overparenthesis on top open vertical bar stack straight s subscript 2 with rightwards arrow on top close vertical bar equals square root of 16 over 9 minus 16 end root rightwards double arrow 4 over 3 square root of 10 equals 4.21 space ft end style

(c) From the centre to the hole.

   Let draw a line OC which is the total displacement from cente to the hole.

    begin mathsize 14px style stack straight s subscript 3 with rightwards arrow on top equals stack straight s subscript 1 with rightwards arrow on top plus stack straight s subscript 2 with rightwards arrow on top stack straight s subscript 3 with rightwards arrow on top equals 2 over 3 straight i with overparenthesis on top plus 2 straight j with overparenthesis on top plus 4 over 3 straight i with overparenthesis on top minus 4 straight j with overparenthesis on top stack straight s subscript 3 with rightwards arrow on top space equals 2 straight i with hat on top plus 2 straight j with hat on top open vertical bar stack straight s subscript 3 with rightwards arrow on top close vertical bar equals 2 square root of 2 equals 2.82 space ft end style

Answer 2

(a) The displacement of the queen’s magnitude from the centre to the front edge is $\frac{2}{3} \sqrt{10} \mathrm{ft}$ .

(b) The displacement of the queen’s magnitude from the front edge to the hole is $\frac{4}{3} \sqrt{10} \mathrm{ft}$ .  

(c) The displacement of the queen’s magnitude from the centre to the hole is $2 \sqrt{2} \mathrm{ft}$  

Explanation:

Let us assume that as shown in the figure, the queen is first placed at point A. So, AB has x ft.

In $\triangle \mathrm{ABC}$ we obtain:

$\tan \theta=x / 2$  … equation (i)

Also, in $\triangle \mathrm{DCE}$, we have:

$\tan \theta=2-x 4$… equation (ii)

From both the equations, we obtain:

$x 2=2-x / 4 \Rightarrow 6 x=4$

$\Rightarrow x=2 / 3 \text { feet. }$

(a) In $\triangle \mathrm{ABC}, \mathrm{AC}=\mathrm{AB} 2+\mathrm{BC} 2$

$=232+22=2 / 3 \sqrt{10} \mathrm{ft}$

(b) In $\Delta \mathrm{CDE}, \mathrm{DE}=2-23=4 / 3 \mathrm{ft}$

$\mathrm{CD}=4 \mathrm{ft}$

Thus, $\mathrm{CE}=\mathrm{CD} 2+\mathrm{DE} 2=42+432=4 / 3$$\sqrt{10}$ $f t$

(c) In $\triangle \mathrm{AGE}, \mathrm{AE}=\mathrm{AG} 2+\mathrm{GE} 2$

$=22+22=8+2$$\sqrt{2}$$f t$