Conservation of Energy
The Physics Hypertextbook™
© 1998-2008 by Glenn Elert -- A Work in Progress
All Rights Reserved -- Fair Use Encouraged
Discussion
The law of conservation of energy cannot be derived. Energy is not concrete; it is not a material substance; it is given meaning through the calculation of numbers. A true historical discussion of the law of conservation of energy is best left to the chapters in this book devoted to thermodynamics. The Nineteenth Century law of conservation of energy of Mayer and von Helmholtz (and Clausius, Carnot, Rumford, Kelvin, and maybe some more). For now, just take it as a really good idea.
The term energy was introduced in the 1850s by William Thomson (Lord Kelvin) and William Rankine. (Really?)
Summary
- bullet
Problems
practice
- The diagram below shows a 10,000 kg bus traveling on a straight road which rises and falls. The horizontal
dimension has been foreshortened. The speed of the bus at point A is 26.82 m/s
(60 mph). The engine has been disengaged and the bus is coasting. Friction
and air resistance are assumed negligible. The numbers on the left show the
altitude above sea level in meters. The letters A-F correspond to points
on the road at these altitudes.
[magnify] - Find the speed of the bus at point B.
- An extortionist has planted a bomb on the bus. If the speed of the bus falls below 22.35 m/s (50 mph) the bomb will explode. Will the speed of the bus fall below this value and explode? If you feel the bus will explode, identify the interval in which this occurs.
- Derive a formula to determine the speed of the bus at any altitude.
Solutions …
- Answer it.
- Two 64 kg stick figures are performing a stunt as the diagram to the right shows. One stick figure stands atop an 7.0 m escarpment. A second stick figure stands on a light, inflexible beam balanced over large rock. The first stick figure rolls a 256 kg boulder off the edge of the escarpment towards the iron beam. (Warning: These are professional stunt stick figures. Don't try this at home.)
[magnify]- What is the theoretical limit on the height to which the second stick figure can rise? Assume that stick figures obey the law of conservation of energy.
- Describe how the design of this stunt alone will prevent the second stick figure from reaching the theoretical limit. Friction, air resistance, and the likelihood that either the beam or the stick figure will break are minor factors.
- Answer it.
- Write something different.
- Answer it.
- Write something completely different.
- Answer it.
conceptual
- Four identical balls are thrown from the top of a cliff, each with
the same speed. The first is thrown straight
up, the second is thrown at 30° above the horizontal, the third at 30°
below he horizontal, and the fourth straight down. How do the
speeds and kinetic energies of the balls compare as they strike the ground …
- when air resistance is negligible?
- when air resistance is significant?
numerical
- A 940 W motor is used to lift 200 kg of supplies 11 m
above street level to the roof of a building.
- If the motor ran for 24 s how much work did it do?
- What is the final potential energy of the supplies relative to street level?
- How much work was done against friction?
- What was the average force of friction on the cable?
- A 45 kg box is pushed up a 21 m ramp at a uniform speed. The top of the ramp is 3.0 m higher than the bottom.
- What is the potential energy of the box at the top of the ramp relative to the bottom of the ramp?
- What work was done pushing the box up the ramp if friction were negligible?
- What work was done pushing the box up the ramp if the force of friction between the box and the ramp was 100. N?
- A 1.2 × 103 kg car driving downhill goes from an altitude of 70 m to 40 m above sea level and accelerates from 11 m/s to 23 m/s.
- How much potential energy did the car lose?
- How much kinetic energy did it gain?
- How much energy is unaccounted for?
- Where did this energy go?
- An 82 kg skydiver jumps from a height of 95 m and strikes
the ground with a speed of 6.0 m/s.
- Calculate the work done by air resistance on the skydiver.
- What was the average air resistance on the skydiver during this jump?
- How does the magnitude of the average air resistance compare to the weight of the skydiver?
- Top pole vaulters have a mass of about 80 kg and can clear a bar 6.0 m above the ground. Top sprinters also have a mass of about 80 kg and can cover 100 m in 10 s. Given these numbers, show that world record pole vaults would not be possible without the pole contributing some elastic potential energy.
statistical
- gold-medal.txt
The accompanying data table gives the men's hundred meter dash and pole vault gold medal results for every summer Olympiad from 1896 to 2000. A crude physical model of the pole vault assumes that all the vaulter's kinetic energy on approach is converted to gravitational potential energy at the top of the vault. As we all know, real world situations are never this simple. If we compare the kinetic energy of a typical world class sprinter to the gravitational potential energy of a similarly accomplished pole vaulter of equal mass, we find that the two numbers are not equal. In the earlier years of the Olympics, energy was lost is the course of the vault; that is, the potential energy of a vaulter was always less than the kinetic energy of a sprinter. (No surprise there. Lost energy is inevitable.) By the end of the Twentieth Century, however, the situation had reversed. The gravitational potential energy of gold medal pole vaulters exceeded the kinetic energy of gold medal sprinters. It would appear that pole vaulters have discovered a way to violate the law of conservation of energy.- Using the accompanying data table, produce a graph that can be used to identify the year in which the maximum gravitational potential energy of Olympic pole vaulters exceeded the average kinetic energy of Olympic sprinters. (Choose an appropriate mass for an athlete and be sure to identify the year of the transition.)
- What changed about the sport that enabled pole vaulters to "violate" the law of conservation of energy? (Was it the shoes? Energy bars? Performance enhancing drugs? No, of course not! Obviously, it has something to do with energy, but you need to be a bit more specific. Where is the extra energy coming from?)
- pile-driver.txt
A group of students performed an experiment driving nails into a wooden block. They used a 1.1091 kg pile driver released at rest from a height h above the block. Before the pile driver fell, the top of the nail was a height s1 above the block. After the pile driver fell, the top of the nail was a height s2 above the block. They repeated the experiment eight times -- four times driving the nail with the grain of the wood and four times driving the nail across the grain. For each trial determine …- the work done by the falling pile driver
- the average force exerted by the pile driver on the nail
- the average acceleration of the pile driver while in contact with the nail
- the speed of the pile driver on impact
- the duration of each impact (in milliseconds), and
- the power of each impact (in kilowatts)
Nailing with the Grain h (m) s1 (m) s2 (m) W (J) F (N) a (m/s2) v (m/s) Δt (ms) P (kW) 0.3060 0.07160 0.06600 0.6115 0.06600 0.05675 0.9180 0.05675 0.04435 1.2220 0.04435 0.03060 Nailing across the Grain h (m) s1 (m) s2 (m) W (J) F (N) a (m/s2) v (m/s) Δt (ms) P (kW) 0.3060 0.06935 0.06670 0.6115 0.06670 0.06095 0.9180 0.06095 0.05370 1.2220 0.05370 0.04640
- the effect that the force of a pile driver has on the distance through which a nail moves for this type of wood.
Resources
- historical
- On the Conservation of Force [Energy], Hermann Helmholtz (1863), Modern History Sourcebook
- The Mechanical Universe and Beyond (video on demand, login
required)
- Conservation of Energy, According to one of the major laws of physics, energy is neither created nor destroyed.
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