Ch 08: Dynamics II: Motion in a Plane

Knight Calc - Physics for Scientists and Engineers 5th Edition

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Knight Calc 5th Edition

Ch 08: Dynamics II: Motion in a Plane

Problem 33

Chapter 8, Problem 33

A 250 g ball is launched with a speed of 35 m/s at a 30° angle. A strong headwind exerts a constant horizontal drag force on the ball. What is the magnitude of the drag force if the wind reduces the ball's travel distance by 20%?

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Convert the mass of the ball from grams to kilograms: \( m = 250 \, \text{g} = 0.250 \, \text{kg} \).

Determine the initial horizontal velocity of the ball using the formula \( v_{x} = v \cdot \cos(\theta) \), where \( v = 35 \, \text{m/s} \) and \( \theta = 30^\circ \).

Calculate the ideal range of the projectile (without drag) using the formula \( R = \frac{v^2 \cdot \sin(2\theta)}{g} \), where \( g = 9.8 \, \text{m/s}^2 \).

Account for the 20% reduction in range due to the drag force. The actual range is \( R_{\text{actual}} = 0.8 \cdot R \).

Use the work-energy principle to relate the work done by the drag force to the reduction in horizontal kinetic energy. The drag force \( F_{\text{drag}} \) can be found using \( W = F_{\text{drag}} \cdot R_{\text{actual}} \), where \( W \) is the work done to reduce the range.

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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 and any other forces acting on it, such as drag. The trajectory of the projectile is typically parabolic, and its motion can be analyzed in two dimensions: horizontal and vertical. Understanding the initial velocity, launch angle, and the effects of gravity is crucial for predicting the object's path and range.

Drag Force

Drag force is the resistance experienced by an object moving through a fluid, such as air. It depends on several factors, including the object's speed, cross-sectional area, and the properties of the fluid. In this scenario, the drag force acts against the ball's motion, reducing its horizontal distance traveled. The magnitude of the drag force can be calculated using the drag equation, which relates it to the velocity and other characteristics of the object.

Distance Reduction

Distance reduction in projectile motion due to external forces, like drag, indicates that the actual range of the projectile is less than what would be expected in a vacuum. In this case, the problem states that the wind reduces the ball's travel distance by 20%. This reduction can be quantified to find the effective range and subsequently used to determine the magnitude of the drag force acting on the ball during its flight.

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