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Exponential functions
Published in John Bird, Bird's Basic Engineering Mathematics, 2021
Exponential functions are used in engineering, physics, biology and economics. There are many quantities that grow exponentially; some examples are population, compound interest and charge in a capacitor. With exponential growth, the rate of growth increases as time increases. We also have exponential decay; some examples are radioactive decay, atmospheric pressure, Newton's law of cooling and linear expansion. Understanding and using exponential functions is important in many branches of engineering.
Nuclear Fuels, Nuclear Structure, the Mass Defect, and Radioactive Decay
Published in Robert E. Masterson, Introduction to Nuclear Reactor Physics, 2017
Shortly after radioactivity was discovered, people began to try to calculate the time it would take for a particular radioactive material to decay. All radioactive elements decay according to a simple, universal exponential decay law that we will study in great detail in this chapter, and in later sections of this book. We will even do a simple experiment to convince you of the authenticity of this law. Mathematically speaking, an exponential decay law is one where the amount of material ΔN lost from a sample of N atoms after a period of time ΔT is proportional to the amount of material that existed before the time the decay occurred. In other words, the fraction of the atoms that decay is the same whether you have a pound of radioactive material or just an ounce. The decay fraction is independent of the amount of atoms in a sample. On the other hand, the number of atoms that decay is directly proportional to how many atoms you start with. The more atoms there are in the sample, the more atoms there are that will decay.
Exponential functions
Published in John Bird, Basic Engineering Mathematics, 2017
Exponential functions are used in engineering, physics, biology and economics. There are many quantities that grow exponentially; some examples are population, compound interest and charge in a capacitor. With exponential growth, the rate of growth increases as time increases. We also have exponential decay; some examples are radioactive decay, atmospheric pressure, Newton’s law of cooling and linear expansion. Understanding and using exponential functions is important in many branches of engineering.
The anaerobic power reserve and its applicability in professional road cycling
Published in Journal of Sports Sciences, 2019
Dajo Sanders, Mathieu Heijboer
The anaerobic reserve could be useful in individualising exercise intensity for HIT. In cycling, the anaerobic power reserve (APR) is defined as the difference between maximal sprint peak power output and power output at VO2max (Weyand et al., 2006). Studies have used the anaerobic reserve range to set out the minimal and maximal values of a short-duration power-duration curve (Bundle, Hoyt, & Weyand, 2003; Sanders et al., 2017; Weyand & Bundle, 2005; Weyand et al., 2006). Subsequently, an exponential decay model is used to describe the decrement in power output over time. The obtained power-duration curve can be used to predict power output over all-outs efforts lasting from a few seconds to a few minutes (Sanders et al., 2017; Weyand et al., 2006). This is a similar approach to the critical power (CP) model (Poole, Burnley, Vanhatalo, Rossiter, & Jones, 2016), however, the CP model mainly applies to longer duration performances (approximately 3 – 45 min) while the APR model focuses on short-duration performance (5 – 300 s). The APR model is based on the assumption that the decrement in (cycling) power output over time in humans, irrespective of between-individual differences in absolute power outputs, is the same when this is expressed in relation to their anaerobic reserve (Bundle et al., 2003; Weyand et al., 2006).