Assertion A: Two identical balls A and B thrown with same velocity ‘u’ at two angles with horizontal attained the same range R

Question

Given below are two statements. One is labeled as Assertion A and the other is labeled as Reason R.

Assertion A: Two identical balls A and B thrown with same velocity ‘u’ at two angles with horizontal attained the same range R. If A and B reached the maximum height $h_{1}$ and $h_{2}$ respectively, then $R = 4 \sqrt{h_{1} h_{2}}$

Reason R: product of heights. $h_{1} h_{2} = (\frac{u^{2} \sin^{2} θ}{2 g}) . (\frac{u^{2} \cos^{2} θ}{2 g})$

Choose the CORRECT answer:

Smart Achievers · Accepted Answer

(1)

$We know that if range is same for two angle of projection, then these angle must be complementary.$

$Let first angle of projection be θ then second will be (90 - θ)$

$∴$ $h_{1} = \frac{u^{2} \sin^{2} θ}{2 g} and h_{2} = \frac{u^{2} \sin^{2} (90 - θ)}{2 g} = \frac{u^{2} \cos^{2} θ}{2 g}$

$∴$ $h_{1} h_{2} = \frac{u^{2} \sin^{2} θ}{2 g} \cdot \frac{u^{2} \cos^{2} θ}{2 g}$

$So, reason is correct$

$\Rightarrow \sqrt{h_{1} h_{2}} = \frac{u^{2} \sin θ \cos θ}{2 g}$

$\Rightarrow 4 \sqrt{h_{1} h_{2}} = \frac{4 u^{2} \sin θ \cos θ}{2 g}$

$\Rightarrow 4 \sqrt{h_{1} h_{2}} = \frac{u^{2} (\sin θ \cos θ)}{g}$

$\Rightarrow 4 \sqrt{h_{1} h_{2}} = \frac{u^{2} \sin 2 θ}{g} = R$

$So, assertion is correct and reason is correct explanation of assertion.$

Solution

1 Both A and R are true and R is the correct explanation of A

2 Both A and R are true and R is NOT the correct explanation of A

3 A is true but R is false

4 A is false but R is true

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