A shell is fired from a fixed artillery gun with an initial speed $u$ such that it hits the target on the ground at a distance R from it. If $\mathrm{t}_{1}$ and $\mathrm{t}_{2}$ are the values of the time taken by it to hit the target in two possible ways, the product $\mathrm{t}_{1} \mathrm{t}_{2}$ is:

(A) $2 \mathrm{R} / \mathrm{g}$

(B) $\mathrm{R} / 4 \mathrm{~g}$

(D) $\mathrm{R} / 2 \mathrm{~g}$

JEE Main · 2019 · 12 Apr 2019 · Shift 1 · Kinematics

A shell is fired from a fixed artillery gun with an initial speed

u

such that it hits the target on the ground at a distance R from it. If

t_{1}

and

t_{2}

are the values of the time taken by it to hit the target in two possible ways, the product

t_{1} t_{2}

is:

(A) $2 R / g$ ✓ Correct
(B) $R /4 g$
(C) $R / g$
(D) $R /2 g$

Answer: (A)

Solution

A fixed speed

u

can reach the same range

R

by two complementary launch angles,

θ

and

9 0^{\circ} - θ

. Their times of flight are

t_{1} = \frac{2 u sin θ}{g}

and

t_{2} = \frac{2 u cos θ}{g}

. Multiplying,

t_{1} t_{2} = \frac{4 u ^{2} sin θ cos θ}{g ^{2}} = \frac{2 u ^{2} sin 2 θ}{g ^{2}}

. Recognising the range formula

R = \frac{u ^{2} sin 2 θ}{g}

inside this expression, we get

t_{1} t_{2} = \frac{2 R}{g}

JEE Main · 2019 · 12 Apr 2019 · Shift 1 · Single correct option

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