Question

The ranges and heights for two projectiles projected with the same initial velocity at angles 42° and 48° with the horizontal are \( R_1, R_2 \) and \( H_1, H_2 \) respectively. Choose the correct option:

• A. \( R_1>R_2 \) and \( H_1 = H_2 \)
• B. \( R_1 = R_2 \) and \( H_1<H_2 \)
• C. \( R_1<R_2 \) and \( H_1<H_2 \)
• D. \( R_1 = R_2 \) and \( H_1 = H_2 \)

Hint:

Any two angles \( \theta \) and \( (90^\circ - \theta) \) will always yield the same horizontal range. However, the steeper the angle, the higher the projectile will travel.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
This question explores the properties of projectile motion, specifically how range and maximum height vary with the angle of projection when initial velocity is constant.

Step 2: Key Formula or Approach:
The formulas for Range (\( R \)) and Maximum Height (\( H \)) are: \[ R = \frac{u^2 \sin(2\theta)}{g} \] \[ H = \frac{u^2 \sin^2\theta}{2g} \]

Step 3: Detailed Explanation:
1. For Range (\( R \)): Range is the same for complementary angles (angles that add up to 90°). Check the angles: \( 42^\circ + 48^\circ = 90^\circ \). Since the angles are complementary, \( \sin(2 \times 42^\circ) = \sin(84^\circ) \) and \( \sin(2 \times 48^\circ) = \sin(96^\circ) \). Using the identity \( \sin(180^\circ - \theta) = \sin\theta \), we see \( \sin(96^\circ) = \sin(180^\circ - 84^\circ) = \sin(84^\circ) \). Therefore, \( R_1 = R_2 \). 2. For Height (\( H \)): Height is proportional to \( \sin^2\theta \). In the range \( 0^\circ \) to \( 90^\circ \), the sine function increases as the angle increases. Since \( 42^\circ<48^\circ \), then \( \sin 42^\circ<\sin 48^\circ \). Consequently, \( H_1<H_2 \).

Step 4: Final Answer
The ranges are equal, but the height is greater for the larger angle. Thus, \( R_1 = R_2 \) and \( H_1<H_2 \).