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

A

1 : 3

B

1 : 1

C

1 : √3

D

√3 : 1

Step-by-Step Solution

  1. Formula for Horizontal Range: The horizontal range RR of a projectile is given by R=v02sin⁡2θ0gR = \frac{v_0^2 \sin 2\theta_0}{g} .
  2. Property of Complementary Angles: The horizontal range is the same for two angles of projection θ\theta and (90∘−θ)(90^{\circ} - \theta) for the same initial speed. This is because sin⁡2θ=sin⁡2(90∘−θ)=sin⁡(180∘−2θ)\sin 2\theta = \sin 2(90^{\circ} - \theta) = \sin(180^{\circ} - 2\theta) .
  3. Application: Given angles are 30∘30^{\circ} and 60∘60^{\circ}. Since 30∘+60∘=90∘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 1:11:1.
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