Question

If two stones A and B are thrown upwards at angles of 30° and 60° respectively with ground at same speed, which stone will have longer horizontal range?

• A. A
• B. B
• C. both A & B will have same range
• D. depend on masses of A and B

Hint:

For projectiles launched with the same speed, the range will be the same for complementary angles \( \theta \) and \( 90^\circ - \theta \).

Answer 1

The Correct Option is C

Step 1: Use the formula for the horizontal range.
The horizontal range \( R \) for a projectile is given by the formula: \[ R = \frac{v^2 \sin(2\theta)}{g}, \] where \( v \) is the velocity, \( \theta \) is the angle of projection, and \( g \) is the acceleration due to gravity.
Step 2: Analyze the situation.
Since the velocity is the same for both stones and the only difference is the angle of projection, the horizontal range depends on \( \sin(2\theta) \). For \( \theta = 30^\circ \), \( \sin(60^\circ) = \frac{\sqrt{3}}{2} \), and for \( \theta = 60^\circ \), \( \sin(120^\circ) = \frac{\sqrt{3}}{2} \). Thus, both stones A and B will have the same range.
Step 3: Conclusion.
Thus, both stones A and B will have the same horizontal range, which corresponds to option (C).

Answer 2

Both stones are thrown at the same speed but at angles of 30° and 60°, and the question asks which has the longer horizontal range. There is a general rule for projectile range: for a fixed launch speed, the ranges at angle \( \theta \) and at angle \( (90^\circ - \theta) \) are always equal, because the range formula depends on \( \sin(2\theta) \), and \( \sin(2(90^\circ-\theta)) = \sin(180^\circ - 2\theta) = \sin(2\theta) \). Since \( 30^\circ + 60^\circ = 90^\circ \), these two angles are complementary, so their ranges must be equal regardless of the actual speed value. Let's check each option against this rule.

1. A: If A had the longer range, the complementary-angle symmetry would be broken, which contradicts the general rule for any launch speed.
2. B: Likewise, if B had the longer range, this would also violate the complementary-angle symmetry that always holds when two angles add up to 90°.
3. Both A and B will have the same range: This is exactly what the complementary-angle rule predicts for 30° and 60°, which sum to 90°.
4. Depends on masses of A and B: The projectile range formula \( R = \frac{v^2\sin(2\theta)}{g} \) contains no mass term at all — range depends only on speed, angle, and gravity, so mass has no bearing on the outcome here.

The complementary-angle symmetry confirms both stones travel the same horizontal range.

Therefore, the correct answer is both A and B will have the same range.