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Rotational Work and Energy

Three project...

Question

Three projectiles are thrown all with same speed $u$ but at different angles of projection $(θ_{1} > θ_{2} > θ_{3}$ ) all taken from horizontal. Maximum height attained by projectile (3) is $H$ . Range of (1) & (3) is same & that of (2) is maximum for the given speed.At height $H$ , speeds are $V_{1}, V_{2}$ & $V_{3}$ as shown. Total time of flights are $T_{1}, T_{2}$ & $T_{3}$ . Choose the correct statement(s).

V_{1} > V_{2} > V_{3}

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V_{1} = V_{2} = V_{3}

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T_{1}^{2}, T_{2}^{2}

T_{3}^{2}

are in AP

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θ_{1}, θ_{2},

θ_{3}

are in AP

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Solution

The correct options are
B $θ_{1}, θ_{2},$ & $θ_{3}$ are in AP
C $V_{1} = V_{2} = V_{3}$
D $T_{1}^{2}, T_{2}^{2}$ & $T_{3}^{2}$ are in AP
From work energy theorem we know that work done by all the forces is equal to the change in kinetic energy,
$V_{1} = V_{2} = V_{3}$
$θ_{1} + θ_{2} = 90$ & $θ_{2} = 45$
$\Rightarrow \frac{θ_{1} + θ_{3}}{2} = θ_{2} \Rightarrow θ_{1}, θ_{2}$ & are in AP
$T_{1} = \frac{2 u sin θ_{1}}{g}$
$\Rightarrow T_{2} = \frac{2 u sin 45^{\circ}}{g} = T_{3} = \frac{2 u cos θ_{1}}{g}$
$\Rightarrow \frac{T_{1}^{2} + T_{3}^{2}}{2} = T_{2}^{2} \Rightarrow T_{1}^{2}, T_{2}^{2}$ & $T_{3}^{2}$ are in AP.
Hence B, C and D are correct options.

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