According to the work-energy theorem, the amount of work done can be determined using which formula?
A. [tex]$W=\Delta K E=2 m\left(v_i^2-v_f^2\right)$[/tex]
B. [tex]$W=\Delta K E=\frac{1}{2} m\left(v_i^2-v_f^2\right)$[/tex]
C. [tex]$W=\Delta K E=2 m\left(v_f^2-v_i^2\right)$[/tex]
D. [tex]$W=\Delta K E=\frac{1}{2} m\left(v_f^2-v_i^2\right)$[/tex]
Answer (2)
The work-energy theorem states that the work done on an object equals its change in kinetic energy.
Kinetic energy is defined as K E = 2 1 m v 2 .
The change in kinetic energy is Δ K E = 2 1 m ( v f 2 − v i 2 ) .
Therefore, the work done is W = Δ K E = 2 1 m ( v f 2 − v i 2 ) .
Explanation
Understanding the Problem The problem asks us to identify the correct formula for the work-energy theorem from a set of options. The work-energy theorem relates the work done on an object to its change in kinetic energy.
Recalling Kinetic Energy The kinetic energy of an object is given by the formula: K E = 2 1 m v 2 where m is the mass and v is the velocity of the object.
Calculating Change in Kinetic Energy The change in kinetic energy, Δ K E , is the difference between the final kinetic energy ( K E f ) and the initial kinetic energy ( K E i ): Δ K E = K E f − K E i = 2 1 m v f 2 − 2 1 m v i 2 where v f is the final velocity and v i is the initial velocity.
Applying the Work-Energy Theorem We can factor out the common term 2 1 m from the expression for Δ K E :
Δ K E = 2 1 m ( v f 2 − v i 2 ) According to the work-energy theorem, the work done, W , is equal to the change in kinetic energy, Δ K E . Therefore: W = Δ K E = 2 1 m ( v f 2 − v i 2 )
Identifying the Correct Formula Comparing this derived formula with the given options, we can identify the correct formula.
Final Answer The correct formula for the work-energy theorem is: W = Δ K E = 2 1 m ( v f 2 − v i 2 ) So the answer is: W = Δ K E = 2 1 m ( v f 2 − v i 2 )
Examples
The work-energy theorem is a fundamental principle in physics that connects the work done on an object to its change in kinetic energy. For example, when designing a roller coaster, engineers use the work-energy theorem to calculate how much energy is required to lift the cars to the top of the first hill, and how fast the cars will be moving at various points along the track. Similarly, in sports, understanding the work-energy theorem can help athletes optimize their performance. For instance, a baseball player can use this principle to understand how the work they do when swinging the bat translates into the kinetic energy of the ball, and ultimately, the distance the ball travels.
The work-energy theorem dictates that the work done is equal to the change in kinetic energy, formulated as W = Δ K E = 2 1 m ( v f 2 − v i 2 ) . Therefore, the correct option is D.
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