Question 77: Motion in 2D & Vectors

Question

Given below are two statements. One is labelled as Assertion A and the other is labelled as Reason R.
Assertion A : Two identical balls A and B thrown with same velocity 'u' at two different angles with horizontal attained the same range R. IF A and B reached the maximum height h1 and h2 respectively, then $$R = 4\sqrt {{h_1}{h_2}} $$
Reason R : Product of said heights.
$${h_1}{h_2} = \left( {{{{u^2}{{\sin }^2}	heta } \over {2g}}} ight)\,.\,\left( {{{{u^2}{{\cos }^2}	heta } \over {2g}}} ight)$$
Choose the correct answer :

Accepted Answer

When two projectiles are thrown with the same initial velocity 'u' but at complementary angles (say, $	heta$ and $(90^\circ - 	heta)$) with the horizontal, they attain the same range R. The formula for the range R of a projectile is:
$$ R = \frac{u^2 \sin 2	heta}{g} $$
For complementary angles, $2 	heta$ and $180^\circ - 2 	heta$ (which simplifies to the same value for the sine function), the ranges are equal.
The maximum height $h_1$ for angle $	heta$ is given by:
$$ h_1 = \frac{u^2 \sin^2 	heta}{2g} $$
And the maximum height $h_2$ for angle $(90^\circ - 	heta)$ is given by:
$$ h_2 = \frac{u^2 \cos^2 	heta}{2g} $$
Now, multiplying these heights:
$$ h_1 h_2 = \left( \frac{u^2 \sin^2 	heta}{2g} ight) \cdot \left( \frac{u^2 \cos^2 	heta}{2g} ight)$$
This simplifies to:
$$ h_1 h_2 = \frac{u^4 \sin^2 	heta \cos^2 	heta}{4g^2}$$
Since $\sin^2 	heta \cos^2 	heta = \left( \frac{\sin 2 	heta}{2} ight)^2 = \frac{1}{4} \sin^2 2 	heta$:
$$ h_1 h_2 = \frac{u^4 \sin^2 2 	heta}{16g^2}$$
Using the range formula $ R = \frac{u^2 \sin 2 	heta}{g} $, we get:
$$ R^2 = \left( \frac{u^2 \sin 2 	heta}{g} ight)^2$$
Thus, we have:
$$ 4h_1 h_2 = \frac{u^4 \sin^2 2 	heta}{4g^2} = \frac{R^2}{4} $$
This simplifies to:
$$ R = 4\sqrt{h_1 h_2}$$
Both the assertion and the reason are correct, and the reason correctly explains the assertion.
The correct answer is:
Option A :
Both A and R are true and R is the correct explanation of A.

Answer

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

Answer

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

Answer

A is true but R is false.

Answer

A is false but R is true.

Motion in 2D & Vectors

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