A 3.5-kg object moving in two dimensions initially has a velocity v1→\overrightarrow{$$v_1$$}_{} = (10.0 î + 20.0 ĵ) m/s. A net force F→\overrightarrow{F} then acts on the object for 2.0 s, after which the object’s velocity is v2→\overrightarrow{$$v_2$$}_{} = (15.0 î + 30.0 ĵ) m/s. Determine the work done by F→\overrightarrow{F} on the object.

Ch. 07 - Work and Energy

Giancoli Douglas - Physics for Scientists and Engineers 5th edition

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Giancoli Douglas 5th edition

Ch. 07 - Work and Energy

Problem 64

Chapter 7, Problem 64

A 3.5-kg object moving in two dimensions initially has a velocity $v sub 1 right arrow sub$ = (10.0 î + 20.0 ĵ) m/s. A net force $\vec{F}$ then acts on the object for 2.0 s, after which the object’s velocity is $v sub 2 right arrow sub$ = (15.0 î + 30.0 ĵ) m/s. Determine the work done by $\vec{F}$ on the object.

Verified step by step guidance

Step 1: Start by calculating the initial kinetic energy (KE₁) of the object using the formula for kinetic energy: KE = (1/2)mv². Here, m = 3.5 kg and the initial velocity vector is v₁→ = (10.0 î + 20.0 ĵ) m/s. First, find the magnitude of v₁→ using the formula |v₁→| = √(v₁x² + v₁y²), where v₁x = 10.0 m/s and v₁y = 20.0 m/s.

Step 2: Similarly, calculate the final kinetic energy (KE₂) of the object using the same formula KE = (1/2)mv². The final velocity vector is v₂→ = (15.0 î + 30.0 ĵ) m/s. Find the magnitude of v₂→ using |v₂→| = √(v₂x² + v₂y²), where v₂x = 15.0 m/s and v₂y = 30.0 m/s.

Step 3: Determine the change in kinetic energy (ΔKE) of the object. The work-energy theorem states that the net work done on an object is equal to the change in its kinetic energy. Use the formula ΔKE = KE₂ - KE₁.

Step 4: Substitute the values of KE₁ and KE₂ into the formula ΔKE = KE₂ - KE₁ to find the work done by the net force F→ on the object. This value represents the total work done.

Step 5: Verify the units of your calculations to ensure consistency. The work done should be expressed in joules (J), as it is the standard unit of work in the SI system.

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Key Concepts

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

Work-Energy Principle

The Work-Energy Principle states that the work done on an object is equal to the change in its kinetic energy. This principle is fundamental in physics as it connects the forces acting on an object to its motion. In this scenario, the work done by the net force can be calculated by finding the difference in kinetic energy before and after the force acts on the object.

Kinetic Energy

Kinetic energy is the energy an object possesses due to its motion, calculated using the formula KE = 0.5 * m * v², where m is the mass and v is the velocity. In this problem, the initial and final velocities of the object are given, allowing us to compute the initial and final kinetic energies. The difference between these values will provide the work done by the net force.

Newton's Second Law

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, expressed as F = m * a. This law is crucial for understanding how the net force affects the object's motion. In this case, while the question focuses on work, understanding the force's role in changing the object's velocity is essential for a complete analysis.

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