A 45 g bug is hovering in the air. A gust of wind exerts a force F⃗=(4.0ı^−6.0ȷ^)×10−2 N\vec{F} = (4.0\hat{\imath} - 6.0\hat{\jmath}) \times $$10^{-2}$$ \, \text{N} on the bug. What is the bug's speed at the end of this displacement? Assume that the speed is due entirely to the wind.

Convert the mass of the bug from grams to kilograms, as SI units require mass in kilograms. Use the conversion: 1 g = 0.001 kg. Thus, the mass of the bug is m = 45 g × 0.001 kg/g. Determine the acceleration of the bug using Newton's second law: F = ma. Rearrange to find acceleration: a = F/m. The force vector is given as F = (4.0 ext{i} - 6.0 ext{j}) × $$10^{-2} N$$, and the mass is m (calculated in step 1). Perform vector division to find the acceleration components. Calculate the magnitude of the acceleration vector using the formula: |a| = ext{sqrt}(a_$$x^2 + a$$_$$y^2$$), where a_x and a_y are the components of the acceleration vector found in step 2. Use the kinematic equation to find the final speed of the bug: v = a imes t, where a is the magnitude of the acceleration (from step 3) and t is the time of displacement. If the time is not provided, it must be assumed or given in the problem context. Combine the results to express the final speed of the bug. Ensure that the speed is due entirely to the wind, as stated in the problem, and verify that all units are consistent (meters per second for speed).

Ch 09: Work and Kinetic Energy

Knight Calc - Physics for Scientists and Engineers 5th Edition

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Knight Calc 5th Edition

Ch 09: Work and Kinetic Energy

Problem 16b

Chapter 9, Problem 16b

A 45 g bug is hovering in the air. A gust of wind exerts a force $\vec{F} = (4.0 \hat{ı} - 6.0 \hat{ȷ}) \times 10^{- 2} N$ on the bug. What is the bug's speed at the end of this displacement? Assume that the speed is due entirely to the wind.

Verified step by step guidance

Convert the mass of the bug from grams to kilograms, as SI units require mass in kilograms. Use the conversion: 1 g = 0.001 kg. Thus, the mass of the bug is m = 45 g × 0.001 kg/g.

Determine the acceleration of the bug using Newton's second law:

F = ma

. Rearrange to find acceleration:

a = F/m

. The force vector is given as

F = (4.0\(\text{i}\) - 6.0\(\text{j}\)) × 10^{-2}

N, and the mass is m (calculated in step 1). Perform vector division to find the acceleration components.

Calculate the magnitude of the acceleration vector using the formula:

|a| = \(\text{sqrt}\)(a_x^2 + a_y^2)

, where

a_x

and

a_y

are the components of the acceleration vector found in step 2.

Use the kinematic equation to find the final speed of the bug:

v = a 	imes t

, where

a

is the magnitude of the acceleration (from step 3) and

t

is the time of displacement. If the time is not provided, it must be assumed or given in the problem context.

Combine the results to express the final speed of the bug. Ensure that the speed is due entirely to the wind, as stated in the problem, and verify that all units are consistent (meters per second for speed).

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Key Concepts

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

Newton's Second Law of Motion

Newton's Second Law states that the acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass. This relationship is expressed by the equation F = ma, where F is the net force, m is the mass, and a is the acceleration. In this scenario, the force exerted by the wind will cause the bug to accelerate, which is essential for determining its final speed.

Kinematics

Kinematics is the branch of mechanics that deals with the motion of objects without considering the forces that cause the motion. It involves concepts such as displacement, velocity, and acceleration. To find the bug's speed at the end of the displacement, we can use kinematic equations that relate these quantities, particularly when the initial speed is zero.

Vector Addition

Vector addition is the process of combining two or more vectors to determine a resultant vector. In this problem, the force exerted by the wind is given as a vector, and understanding how to add vectors is crucial for analyzing the bug's motion. The components of the force vector will influence the bug's acceleration in both the x and y directions, affecting its overall speed.

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