Kinetic Energy

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© 1998-2008 by Glenn Elert -- A Work in Progress
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Discussion

Derivation when acceleration is constant. Start from the work-energy theorem.

ΔK =  W
ΔK =  FΔs = maΔs

Substituting the value of acceleration determined from the velocity-displacement equation of motion.

ΔK = m  ⎛
⎜
⎝		 v2 − v02 ⎞
⎟
⎠		  Δs
2Δs

Thus

ΔK =  1  mv2 −  1  mv02
2 2

Derivation using calculus. Again, start from the work-energy theorem

ΔK = W  ⌠
⌡		 F(r) · dr = m a · dr = m ⌠
⌡		 dv  · dr
dt

Rearrange the differential terms to get the integral and the function into agreement.

ΔK = m ⌠
⌡		 dv  · dr = m  ⌠
⌡		 dr  · dv = m  ⌠
⌡		  v · dv
dt dt

The integral of which is quite simple to evaluate.

ΔK =  1  mv2 −  1  mv02
2 2

Naturally, the kinetic energy of an object at rest should be zero. Thus an object's kinetic energy is defined mathematically by the following equation …

K =  1  mv2
2

Thomas Young (1773-1829) derived a similar formula in 1807, although he neglected to add the ½ to the front and he didn't use the words mass and weight with the same precision we do nowadays. He was also the first to use the word energy (with its current meaning) in a lecture on collisions given before the Royal Institution. He was the first to use the term ‘energy' for the product of the mass of a body into the square of its velocity, and the expression ‘labour expended' (work done) for the product of the force exerted on a body into the distance through which it is moved, and to state that these two products were proportional to each other.

The term energy may be applied, with great propriety, to the product of the mass or weight of a body, into the square of the number expressing its velocity.

Thomas Young. "Lecture 8." Course of Lectures on Natural Philosophy, 1807.

Young just called it energy. Lord Kelvin (1824-1907) added the adjective "kinetic" to separate it from "potential energy", which was invented by William Rankine (1820-1872) in 1853.

Summary

Problems

practice

  1. A supertanker doesn't come with brakes. Using engines alone, it takes a loaded supertanker 13 km (8 miles) to stop. A typical vessel of this class has a gross mass of about 150 million kilograms and a cruising speed of 50 kph (30 mph). Determine …
    1. the average stopping force applied to the ship and
    2. the average power dissipated by stopping it.

    Solutions …

    1. This problem lends itself well to standard techniques. State the given quantities and convert them to SI units as needed. Use the Work-Energy Theorem. The change in the supertanker's kinetic energy is due to the work done by its engines running in reverse. (It can also be solved using Newton's Second Law of motion -- as it was an earlier section of this book. Both methods lead to exactly the same answer -- as they should.)
                         
      Δs =  13 km = 13,000 m W =  ΔK
      m =  1.50 × 106 kg FΔs =  1  mv02
      v =  0 m/s 2
      v0 =  50 km   1000 m  = 13.888.… m/s F =  mv02  =  (1.50 × 106 kg)(13.888.… m/s)2
      1 h 3600 s 2Δs 2(13,000 m)
                F =  1,112,892… N = 1.1 MN
    2. Power is the rate at which work is done, by definition, but it is also the product of force and velocity; that is, the product of the average force (which we've just calculated) and the average velocity (which can be calculated quite simply).
               
      Pave =  Favevave = 1,112,892.… N  ⎛
      ⎝      		 13.888… m/s + 0 m/s ⎞
      ⎠
      2
      Pave =  7,728,415.… W = 7.7 MW
               
  2. Write something else.
    • Answer it.
  3. Write something different.
    • Answer it.
  4. Write something completely different.
    • Answer it.

numerical

  1. The Enhanced Fujita Scale is a system implemented by the National Weather Service in the US to rank the intensity of tornadoes. EF Scale values are assigned based on three second wind gust speeds. These speeds are estimated (not measured) from observed dammage. EF Scale numbers are often reported by the media.
     
    Operational Enhanced Fujita Scale
      three second gust  
    scale (mph) (m/s) typical damage
    EF0 65-85 29-38 Light:
    Some damage to chimneys; branches broken off trees; shallow-rooted trees pushed over; sign boards damaged.
    EF1 86-110 38-49 Moderate:
    Peels surface off roofs; mobile homes pushed off foundations or overturned; moving autos blown off roads.
    EF2 111-135 50-60 Considerable:
    Roofs torn off frame houses; mobile homes demolished; boxcars overturned; large trees snapped or uprooted; light-object missiles generated; cars lifted off ground.
    EF3 136-165 60-74 Severe:
    Roofs and some walls torn off well-constructed houses; trains overturned; most trees in forest uprooted; heavy cars lifted off the ground and thrown.
    EF4 166-200 75-89 Devastating:
    Well-constructed houses leveled; structures with weak foundations blown away some distance; cars thrown and large missiles generated.
    EF5 > 200 > 200 Incredible:
    Strong frame houses leveled off foundations and swept away; automobile-sized missiles fly through the air in excess of 100 meters; trees debarked; incredible phenomena will occur.
    Source: Online Tornado FAQ, Storm Prediction Center, National Weather Service
     
    How many times more intense is …
    1. an EF2 than an EF1 tornado,
    2. an EF5 than an EF4 tornado,
    3. an EF5 than an EF1 tornado?

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