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: For stone A (30°): horizontal speed \( v\cos30^\circ = \frac{v\sqrt{3}}{2} \), time of flight \( t_A = \frac{2v\sin30^\circ}{g} = \frac{v}{g} \).
Step 2: For stone B (60°): horizontal speed \( v\cos60^\circ = \frac{v}{2} \), time of flight \( t_B = \frac{2v\sin60^\circ}{g} = \frac{v\sqrt{3}}{g} \).
Step 3: Range = horizontal speed × time of flight: \( R_A = \frac{v\sqrt{3}}{2}\cdot\frac{v}{g} = \frac{v^2\sqrt{3}}{2g} \), and \( R_B = \frac{v}{2}\cdot\frac{v\sqrt{3}}{g} = \frac{v^2\sqrt{3}}{2g} \) — the two ranges come out identical.
\[ \boxed{R_A = R_B} \]

Answer 2

We can also just plug in concrete numbers to check. Take \( v = 10 \, \text{m/s} \) and \( g = 10 \, \text{m/s}^2 \). For stone A at 30°: \( R_A = \frac{v^2\sin(60^\circ)}{g} = \frac{100 \times 0.866}{10} \approx 8.66 \, \text{m} \). For stone B at 60°: \( R_B = \frac{v^2\sin(120^\circ)}{g} = \frac{100 \times 0.866}{10} \approx 8.66 \, \text{m} \). Let's check each option against these computed numbers.

1. A: The computed range for A (8.66 m) is not larger than B's, so A does not have the longer range.
2. B: Likewise, B's computed range (8.66 m) is not larger than A's, so B does not have the longer range either.
3. Both A and B will have the same range: The two numeric values computed above are identical (8.66 m each), confirming this option directly.
4. Depends on masses of A and B: No mass value was needed anywhere in this calculation — only speed, angle, and gravity — so the range cannot depend on mass.

The numeric check confirms both stones cover exactly the same horizontal range.

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