To solve this problem, we need to analyze the physics of projectile motion. We have two projectiles launched at the same speed but at different angles, 42° and 48°. We want to compare their ranges (\( R_1 \) and \( R_2 \)) and maximum heights (\( H_1 \) and \( H_2 \)).
- Formula for Range: The range \( R \) of a projectile launched with an initial velocity \( v_0 \) at an angle \( \theta \) is given by: \(R = \frac{v_0^2 \sin 2\theta}{g}\), where \( g \) is the acceleration due to gravity.
- Comparison of Ranges:
- The formula includes \( \sin 2\theta \). For angles 42° and 48°:
- \(\sin 2(42^\circ) = \sin 84^\circ\)
- \(\sin 2(48^\circ) = \sin 96^\circ\)
- Since \(\sin 84^\circ = \sin 96^\circ\) because \(\sin (90^\circ + x) = \cos x\) and \(\cos 6^\circ \approx \sin 84^\circ\), both ranges \( R_1 \) and \( R_2 \) are equal.
- Formula for Maximum Height: The maximum height \( H \) achieved by a projectile is: \(H = \frac{v_0^2 \sin^2 \theta}{2g}\).
- Comparison of Heights:
- The formula depends on \(\sin^2 \theta\).
- \(\sin^2 42^\circ\) is different from \(\sin^2 48^\circ\).
- Since \(\sin 48^\circ > \sin 42^\circ\), it follows that \(\sin^2 48^\circ > \sin^2 42^\circ\), so \( H_2 > H_1 \).
- Conclusion: The ranges are equal and the height of the projectile launched at 48° is greater than that of 42°, leading to the correct option being:
\( R_1 = R_2 \) and \( H_1 < H_2 \)