Two projectiles A and B are thrown with initial velocities of 40 m / s and 60 m / s at angles 30° and 60° with the horizontal respectively. The ratio of their ranges respectively is (g=10 m / s²)

Solution

Physics sir · 4 steps

The angles are complementary, so the sin 2θ parts are equal and only u² decides the ratio.

1. Range formula. For a projectile landing at the same level it was thrown from, the horizontal range depends on the launch speed and on sin 2θ.

   R = (u² sin 2θ)/g

2. Compare the angles. For A, sin 60° = √3/2. For B, sin 120° = √3/2. The two are equal because 30° and 60° add up to 90°. Since g is also the same for both, these factors cancel in the ratio.
3. Ratio of speeds squared. R_A/R_B = u_A²/u_B² = 40²/60² = 1600/3600 = 4/9.
4. So. The ranges are in the ratio 4 : 9, which is option (a). As a check, R_A = 80√3 m and R_B = 180√3 m.

The idea to keep. Complementary angles give equal sin 2θ. With the same speed they give the same range. With different speeds, the range ratio is just the ratio of u².

Answer from the NTA final answer key. Solution by PrepWiser.