Question

Galileo in his book to new sciences started thatfor elevations which exceed for fall short of 45 degree by equal amounts the ranges are equal prove this statement​

Answer 1

Answer:

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Answer 2

In his book on new sciences, Galileo stated that the ranges are equivalent for elevations that surpass and fall short of 45 degrees by equal amounts, i.e    $R_{1} = R_{2}$ hence proved.

Explanation:

When an object is thrown in an upward direction, the body falls to the ground due to gravity's effect. During flight, the body moves in a horizontal direction at a steady speed.

For elevation:

The angle is (45∘+θ).

Then the range is calculated as

$R_{1} =\frac{ V_{0} ^{2}}{g}$sin[2(45+θ)]

here,

$V_{0} = velocity$

g = acceleration due to gravity

$R_{1} = \frac{V_{0} ^{2}}{g}$ Sin [90+2θ]

= $\frac{V_{0^{2} } }{g}$Cos2θ

Sin [90+2θ] = Cos2θ

$R_{2} =\frac{ V_{0} ^{2}}{g}$sin[2(45 - θ)]

$R_{2} =\frac{ V_{0} ^{2}}{g}$ Sin [90 - 2θ]

$R_{2} =\frac{ V_{0} ^{2}}{g}$Cos2θ

Sin [90 - 2θ] =  Cos2θ

From the above equation,

$R_{1} = R_{2}$

This is the statement's necessary proof.

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