Problem 28 - Master Resource Book in JEE Main Physics

Question

Two stones thrown at different angles have same initial velocity and same range. If \(H\) is the maximum height attained by one stone thrown at an angle of \(30^{\circ}\), then the maximum height attained by the other stone is (a) \(\frac{H}{2}\) (b) \(\underline{H}\) (c) \(2 H\) (d) \(3 \mathrm{H}\)

Step-by-Step Solution

Answer

The maximum height attained by the other stone is 3H.

Step 1: Understanding Projectile Motion Range

For two projectiles thrown with the same initial velocity and landing at the same range, the angles of projection must be complementary. This means if one angle is \(30^{\circ}\), the other must be \(90^{\circ} - 30^{\circ} = 60^{\circ}\).

Step 2: Range Equation for Projectile Motion

The range \(R\) of a projectile is given by \( R = \frac{v^2 \sin 2\theta}{g} \), where \(v\) is initial velocity, \(\theta\) is the angle of projection, and \(g\) is the acceleration due to gravity. Since the range is the same for both projectiles, we have: \( v^2 \sin 60^{\circ} = v^2 \sin 30^{\circ} \).

Step 3: Maximum Height of a Projectile

The maximum height \(H\) of a projectile is calculated using \( H = \frac{v^2 \sin^2 \theta}{2g} \). For an angle of \(30^{\circ}\), this height is \(H = \frac{v^2 \sin^2 30^{\circ}}{2g} \).

Step 4: Height Calculation for 60-degree Angle

For the second stone projected at \(60^{\circ}\), use the formula \(H' = \frac{v^2 \sin^2 60^{\circ}}{2g} \). Plug in the values of sine: \(\sin^2 60^{\circ} = \left(\frac{\sqrt{3}}{2}\right)^2 = \frac{3}{4}\), compared to \(\sin^2 30^{\circ} = \left(\frac{1}{2}\right)^2 = \frac{1}{4}\).

Step 5: Comparing Both Heights

For \(30^{\circ}\), \(H = \frac{v^2}{8g}\). For \(60^{\circ}\), \(H' = \frac{3v^2}{8g}\). Thus, \(H' = 3H\). Therefore, the maximum height attained by the second stone is three times that of the first.

Key Concepts

Complementary Angles

In projectile motion, complementary angles are key to understanding how two projectiles can have the same range but different trajectories and maximum heights. Complementary angles, in simple terms, add up to 90 degrees. So, if one projectile is launched at an angle \(\theta\), the other should be at an angle \(90^{\circ} - \theta\). This ensures they land at the same distance when thrown with the same initial speed. A helpful way to imagine this is thinking of a basketball shot. If you shoot from a low angle, say \(30^{\circ}\), and then shoot from a high arcing angle of \(60^{\circ}\), both shots will cover the same horizontal range if thrown with equal strength. The physics behind this is rooted in the range formula where the sine function and complementary angles play a crucial role.

Maximum Height of a Projectile

Achieving the maximum height in projectile motion depends on the angle of launch. For any projectile, this height is given by the formula: \[ H = \frac{v^2 \sin^2 \theta}{2g} \]This formula tells us that the initial velocity \(v\) squared, the sine of the angle squared, and the gravitational pull \(g\) are all part of finding how high the projectile will rise.When we consider a launch angle of \(30^{\circ}\), the sine of this angle is \(0.5\). Therefore, the maximum height reached by the projectile can be calculated using \( H = \frac{v^2 (0.5)^2}{2g} \), resulting in \( H = \frac{v^2}{8g} \).Now, if we take a stone launched at \(60^{\circ}\), the sine value turns into \(\frac{\sqrt{3}}{2}\), which increases the height to \(\frac{3v^2}{8g} \). This high-angle shot provides a much greater maximum height, exactly 3 times higher than the \(30^{\circ}\) launch.

Range of Projectile

The range of a projectile is how far it travels horizontally before hitting the ground. It's determined by several factors such as the initial speed, angle of projection, and gravitational force. The equation that gives us the range is:\[ R = \frac{v^2 \sin 2\theta}{g} \]Here, \(v\) is the initial velocity, \(\theta\) the launch angle, and \(g\) the acceleration due to gravity. An interesting aspect of complementary angles in projectile motion is that they produce the same range.Using our stone example, launching at \(30^{\circ}\) and \(60^{\circ}\) angles with the same initial speed means both will hit the ground at the same horizontal distance. This happens because \( \sin 60^{\circ} = \sin (2 \times 30^{\circ}) \), showing how beautifully mathematics and physics harmonize to keep things consistent across different angles.