In the absence of air resistance, a projectile that lands at the elevation from which it was launched achieves maximum range when launched at a 45° angle. Suppose a projectile of mass m is launched with speed into a headwind that exerts a constant, horizontal retarding force ${\vec{F}}_{wind} = - F_{wind} \hat{ı} .$ By what percentage is the maximum range of a 0.50 kg ball reduced if $F sub wind equals 0.60 cap N$ ?

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Step 1: Begin by understanding the problem. The projectile is launched at an angle of 45° with an initial speed, and there is a horizontal retarding force due to the headwind. The goal is to determine the percentage reduction in the maximum range caused by this force.

Step 2: Write down the equations of motion for the projectile. The horizontal motion is affected by the retarding force, while the vertical motion is governed by gravity. The horizontal acceleration due to the headwind is given by $ a_x = \frac{-F_{\text{wind}}}{m} $, where $ F_{\text{wind}} $ is the force exerted by the wind and $ m $ is the mass of the projectile.

Step 3: Calculate the time of flight of the projectile. The time of flight is determined by the vertical motion. Using the kinematic equation $ t = \frac{2v_0 \sin \theta}{g} $, where $ v_0 $ is the initial speed, $ \theta $ is the launch angle, and $ g $ is the acceleration due to gravity.

Step 4: Determine the horizontal range of the projectile. The range is given by $ R = v_0 \cos \theta \cdot t + \frac{1}{2} a_x t^2 $, where $ a_x $ is the horizontal acceleration due to the wind. Substitute $ a_x = \frac{-F_{\text{wind}}}{m} $ and $ t $ from Step 3 into this equation.

Step 5: Compare the range with and without the wind. The range without the wind is $ R_0 = \frac{v_0^2 \sin 2\theta}{g} $. Calculate the percentage reduction in range using $ \text{Percentage Reduction} = \frac{R_0 - R}{R_0} \times 100 $. Substitute the given values $ m = 0.50 \text{ kg} $, $ F_{\text{wind}} = 0.60 \text{ N} $, and solve symbolically to find the percentage reduction.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Projectile Motion

Projectile motion refers to the motion of an object that is launched into the air and is subject to gravitational force. The trajectory of a projectile is typically parabolic, and its range is influenced by the launch angle, initial speed, and the effects of forces such as gravity and air resistance. In ideal conditions, the maximum range occurs at a 45° launch angle, balancing vertical and horizontal components of motion.

Forces Acting on the Projectile

In this scenario, the projectile is affected by two main forces: gravity, which acts downward, and a horizontal retarding force due to the headwind. The retarding force reduces the horizontal component of the projectile's velocity, thereby affecting its range. Understanding how these forces interact is crucial for calculating the new range of the projectile when subjected to the headwind.

Range Reduction Calculation

To determine the percentage reduction in range due to the headwind, one must first calculate the original range without the wind and then the new range with the wind's effect. The percentage reduction can be found using the formula: ((original range - new range) / original range) * 100%. This calculation highlights the impact of external forces on projectile motion and is essential for solving the given problem.

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Textbook Question

In the absence of air resistance, a projectile that lands at the elevation from which it was launched achieves maximum range when launched at a 45° angle. Suppose a projectile of mass m is launched with speed into a headwind that exerts a constant, horizontal retarding force ${\vec{F}}_{wind} = - F_{wind} \hat{ı} .$ . Find an expression for the angle at which the range is maximum.

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