Problem 4 - Master Resource Book in JEE Main Physics

Question

The speed of projection of a projectile is increased by \(10 \%\), without changing the angle of projection. The percentage increase in the range will be (a) 109 (b) \(20 \%\) (c) \(15 \%\) (d) \(5 \%\)

Step-by-Step Solution

Answer

The percentage increase is approximately 21%, which may suggest an answer choice of around 20% in a practical context.

Step 1: Understand the Range Formula

The range \( R \) of a projectile is given by the formula \( R = \frac{u^2 \sin(2\theta)}{g} \), where \( u \) is the initial speed, \( \theta \) is the angle of projection, and \( g \) is the acceleration due to gravity. This formula helps us understand the relationship between projectile speed and range.

Step 2: Calculate the New Speed

The speed of projection is increased by \(10\%\), meaning the new speed \( u' \) is \( u' = 1.1u \). Here, \( u \) is the original speed of the projectile.

Step 3: Calculate the New Range

Substitute the new speed into the range formula: \( R' = \frac{(1.1u)^2 \sin(2\theta)}{g} = \frac{1.21u^2 \sin(2\theta)}{g} \). Notice \( R' = 1.21R \), where \( R \) is the original range.

Step 4: Determine the Percentage Increase

The percentage increase in range can be calculated as \( \left( \frac{R' - R}{R} \right) \times 100\% = \left( \frac{1.21R - R}{R} \right) \times 100\% = 0.21 \times 100\% = 21\% \).

Step 5: Choose the Correct Option

The calculated percentage increase in range is \(21\%\), which does not directly match any of the provided answers. Verify if any rounding or interpretation might influence this discrepancy. However, considering the closest matching logic, the answer may align with a rounded or standardized choice in practical scenarios.

Key Concepts

Range of a Projectile

Projectile motion is a fascinating topic in physics, and understanding the concept of range is essential for solving many problems. The **range** of a projectile is the horizontal distance it travels during its motion. It depends on several factors, such as the initial speed of the projectile, the angle of projection, and the acceleration due to gravity.

The formula for calculating the range \( R \) of a projectile is given by: \[ R = \frac{u^2 \sin(2\theta)}{g} \] Where:

• \( u \) is the initial speed of the projectile,
• \( \theta \) is the angle of projection,
• and \( g \) is the acceleration due to gravity, typically \( 9.8 \, \text{m/s}^2 \) on Earth.

From this formula, we can observe some key points:

• If you increase the initial speed \( u \), the range \( R \) increases—directly proportional to the square of the speed.
• The sine term \( \sin(2\theta) \) shows that the range depends on the angle of projection in a periodic manner.
• Optimal range is typically achieved at an angle of \( 45^\circ \) in ideal conditions, where air resistance is neglected.

Angle of Projection

The **angle of projection** has a significant impact on the path that a projectile follows and consequently on its range and height. When projecting an object, altering the angle affects how high and how far the projectile will travel.

- **Key Insights:** - **Optimal Angle for Range:** An angle of \(45^\circ\) typically gives the maximum range. This is because \( \sin(90^\circ) \) results in the highest possible value of the sine function, which is 1. - **Lower Angles Provide a Flatter Trajectory:** With a smaller angle, the projectile travels faster horizontally but doesn’t reach as high. - **Higher Angles Offer a Taller Arc:** If the angle is increased beyond \(45^\circ\), the projectile reaches greater heights but the horizontal range diminishes due to the shorter forward velocity component.

- **Real-world Considerations:** In practical scenarios, factors like air resistance and surface angle need to be considered. These factors often require slight adjustments to the ideal angle of \(45^\circ\). For example, throwing a ball in a windy environment may necessitate varying the angle to counteract wind effects.

Effects of Initial Speed on Projectile

In projectile motion, the **initial speed** plays a crucial role in determining the characteristics of the projectile's path, including its range and height. Understanding these effects helps in predicting and manipulating the motion based on different circumstances.

When we increase the initial speed of the projectile:

• **The Range Increases:** The range is directly proportional to the square of the initial speed \( (u^2) \). Therefore, even a small increase in speed leads to a more significant increase in range. For instance, if we increase the speed by \(10\%\), as shown in the step-by-step solution, the range increases by approximately \(21\%\).
• **Trajectories Become More Extended:** Higher speed results in a longer trajectory, meaning the projectile spends more time in the air and covers a greater distance horizontally.
• **Height Achieved Changes:** Although primarily discussed with range, the maximum height reached by the projectile also depends on the initial speed. With greater initial speeds, projectiles can achieve higher altitudes if other conditions like the angle remain unchanged.

Understanding the initial speed’s impact allows engineers and scientists to design better equipment and systems where motion characteristics are critical, such as sports equipment design and spacecraft launch mechanics.