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Zorluk: ZorWork, Energy, and Power

A mechanical test cart accelerates horizontally from rest to a speed of vv, requiring 400 J400\text{ J} of net work. How much additional net work must be performed on the cart to further increase its speed from vv to 3v3v?

  1. A
    1,200 J1,200\text{ J}
  2. B
    3,600 J3,600\text{ J}
  3. 3,200 J3,200\text{ J}Cevap
  4. D
    800 J800\text{ J}

Cevap

An additional 3,200 J3,200\text{ J} of net work is required.
According to the Work-Energy Theorem, the net work performed on an object equals its change in kinetic energy (Wnet=ΔKEW_{\text{net}} = \Delta \text{KE}). Accelerating from rest to speed vv requires 400 J400\text{ J}, meaning 12mv2=400 J\frac{1}{2}mv^2 = 400\text{ J}. At triple the speed (3v3v), the total kinetic energy becomes 12m(3v)2=9(12mv2)=9×400 J=3,600 J\frac{1}{2}m(3v)^2 = 9\left(\frac{1}{2}mv^2\right) = 9 \times 400\text{ J} = 3,600\text{ J}. The additional work needed to increase speed from vv to 3v3v is the difference between these two energy states: 3,600 J−400 J=3,200 J3,600\text{ J} - 400\text{ J} = 3,200\text{ J}.

Adım Adım Çözüm

1
Express the initial kinetic energy and relate it to initial work done.
KE1=12mv2=400 J\text{KE}_1 = \frac{1}{2}mv^2 = 400\text{ J}
By the Work-Energy Theorem, the net work done on an object starting from rest equals its final kinetic energy.
2
Calculate the total kinetic energy of the cart when its speed reaches 3v3v.
KE2=12m(3v)2=9(12mv2)=9×400 J=3,600 J\text{KE}_2 = \frac{1}{2}m(3v)^2 = 9\left(\frac{1}{2}mv^2\right) = 9 \times 400\text{ J} = 3,600\text{ J}
Kinetic energy depends on the square of velocity, so tripling the velocity increases kinetic energy by a factor of 32=93^2 = 9.
3
Determine the additional net work needed to transition from speed vv to 3v3v.
Wadditional=KE2−KE1=3,600 J−400 J=3,200 JW_{\text{additional}} = \text{KE}_2 - \text{KE}_1 = 3,600\text{ J} - 400\text{ J} = 3,200\text{ J}
The additional work required equals the change in kinetic energy between the two states.

Anahtar Kavram

Work-Energy Theorem and Velocity-Squared Relationship of Kinetic Energy
Tahmini Süre:1m 15s
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Work, Energy, and Power

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