hayomi4u
Answered

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If the horizontal range and the max height of a body projected at an angle titre to the horizontal is K and Q respectively. Show that the muzzle velocity Vo is given by: Vo=√[2gQ + K²/8Q]​

Sagot :

Explanation:

Recall that

[tex]K = \dfrac{v_0^2\sin2\theta}{g}\:\:\:\:\:\:\:\:\:(1)[/tex]

and

[tex]Q = \dfrac{v_0^2\sin^2\theta}{2g}\:\:\:\:\:\:\:\:\:(2)[/tex]

From Eqn(2), we can write

[tex]\sin\theta = \sqrt{\dfrac{2gQ}{v_0^2}}\:\:\:\:\:\:\:\:\:(3)[/tex]

Using the identity [tex]\sin\theta = 2\sin\theta \cos\theta[/tex], we can rewrite Eqn(1) as

[tex]\dfrac{gK}{2v_0^2} = \sin\theta \cos\theta[/tex]

Squaring the above equation, we get

[tex]\dfrac{g^2K^2}{4v_0^4} = \sin^2\theta \cos^2\theta[/tex]

[tex]\:\:\:\:\:\:\:\:\:=\sin^2\theta(1 - \sin^2\theta)\:\:\:\:\:\:\:(4)[/tex]

Use Eqn(3) on Eqn(4) and we will get the following:

[tex]\dfrac{g^2K^2}{4v_0^4} = \dfrac{2gQ}{v_0^2}(1 - \dfrac{2gQ}{v_0^2})[/tex]

This simplifies to

[tex]\dfrac{gK^2}{8v_0^2Q} = 1 - \dfrac{2gQ}{v_0^2}[/tex]

Rearranging this further, we get

[tex]1 = \dfrac{2gQ}{v_0^2} + \dfrac{gK^2}{8v_0^2Q}[/tex]

Putting [tex]v_0^2[/tex] to the left side, we get

[tex]v_0^2 = 2qQ + \dfrac{gK^2}{8Q}[/tex]

Finally, taking the square root of the equation above, we get the expression for the muzzle velocity [tex]v_0[/tex] as

[tex]v_0 = \sqrt{2gQ + \dfrac{gK^2}{8Q}}[/tex]