If a body A of mass $M$ is thrown with a velocity $v$ at an angle of $30^{\circ}$ to the horizontal and anothe

Question

If a body A of mass $M$ is thrown with a velocity $v$ at an angle of $30^{\circ}$ to the horizontal and another body B of the same mass is thrown with the same speed at an angle of $60^{\circ}$ to the horizontal. The ratio of horizontal range of A to B will be:

Accepted Answer

1.  **Formula for Horizontal Range:** The horizontal range $R$ of a projectile is given by $R = \frac{v_0^2 \sin 2	heta_0}{g}$ .
2.  **Property of Complementary Angles:** The horizontal range is the same for two angles of projection $	heta$ and $(90^{\circ} - 	heta)$ for the same initial speed. This is because $\sin 2	heta = \sin 2(90^{\circ} - 	heta) = \sin(180^{\circ} - 2	heta)$ .
3.  **Application:** Given angles are $30^{\circ}$ and $60^{\circ}$. Since $30^{\circ} + 60^{\circ} = 90^{\circ}$, these are complementary angles. Therefore, their ranges are equal.
4.  **Mass Independence:** The trajectory and range of a projectile (ignoring air resistance) are independent of the mass of the body . Thus, the ratio of ranges is $1:1$.

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