# Relationship between kinetic energy and amplitude

### Photoelectric effect (article) | Photons | Khan Academy

be the extension of the spring: i.e., the difference between the spring's actual length determines the times at which the oscillation attains its maximum amplitude, $x=a$ The kinetic energy attains its maximum value, and the potential energy. The kinetic energy of photoelectrons remains constant as light amplitude . The relationship between photoelectron kinetic energy and light frequency is shown. and the kinetic energy is Since the total mechanical energy E is equal to the sum of kinetic and This fact allows us to derive some a useful relationship.

Both kinetic energy and momentum are related to the speed of the object. Thus, we can relate the kinetic energy to the momentum as follows: Then, we can use these equations with the wavelength-momentum relation to write the wavelength in terms of the kinetic energy.

We can use this equation for any object. To describe an electron using units of electron volts eV for energy and nanometers nm for wavelength, the equation above becomes We can use this result to find the wavelength of the wave function in each region of potential energy. In general we will not calculate the exact number for the wavelength.

Instead, we will want to know how the wavelength in one region compares to that in another.

### Energy Transport and the Amplitude of a Wave

For that comparison we can use the equation above to note that as the kinetic energy goes up the wavelength goes down and as the kinetic energy goes down the wavelength goes up. Looking at the equation, we can also see that the kinetic energy must be positive for the wavelength to have any meaning.

If the kinetic energy is negative, we get the square root of a negative number. That gives us a meaningless wavelength.

Thus, we can use this result only for positive kinetic energies. When the total energy is greater than the potential energy, determine the wavelength of the wave function by using the relation between the wavelength and the kinetic energy.

Determining Wavelengths -- Example As an example, use the potential energy for an electron entering a metal. Here, we repeat the earlier diagram but include a dotted line to indicate the total energy of the electron. To determine the kinetic energy of the electron we look at the difference between the total energy and the potential energy. As shown in the next figurethe kinetic energy on the left side is about four times greater than that on the right side of the edge of the metal.

So, we conclude that the wavelength on the left side will be about one-half of that on the right side. The approximate wave function looks like the one in this figure.

**Simple Harmonic Motion, Mass Spring System - Amplitude, Frequency, Velocity - Physics Problems**

Wave Functions at Boundaries When drawing wave functions at changes in potential energy, we make a smooth connection between the wave functions in the two regions. This type of connection comes from an understanding that it would be difficult to make sense of different probabilities of finding the electron if we approach the boundary from different directions.

For example, suppose that this wave function could be possible. The probability distribution, given by the square of this wave function, looks like this. If we approach this boundary from the left, we obtain the probability marked as A. However, if we approach from the right we get a different probability B at the boundary. A different value occurs depending on which way we approach the boundary, yet it is the same boundary.

Two different values for the probability at the same point would be meaningless. So, we have another rule, called the smoothness condition, for drawing wave functions: The wave function must be smoothly connected at all points including boundaries where the potential energy changes.

Constraints on the Wave Function: An Infinite Potential Example In many conditions, the physical situation will lead to constraints on the wave function. This is also what physicists believed would happen if the light intensity was increased.

Light amplitude was expected to be proportional to the light energy, so higher amplitude light was predicted to result in photoelectrons with more kinetic energy. Classical physicists also predicted that increasing the frequency of light waves at a constant amplitude would increase the rate of electrons being ejected, and thus increase the measured electric current.

Using our beach ball analogy, we would expect waves hitting the dock more frequently would result in more beach balls being knocked off the dock compared to the same sized waves hitting the dock less often. Now that we know what physicists thought would happen, let's look at what they actually observed experimentally! When experiments were performed to look at the effect of light amplitude and frequency, the following results were observed: The kinetic energy of photoelectrons increases with light frequency.

Electric current remains constant as light frequency increases. Electric current increases with light amplitude. The kinetic energy of photoelectrons remains constant as light amplitude increases.

## Energy Transport and the Amplitude of a Wave

Energy-Amplitude Mathematical Relationship The energy transported by a wave is directly proportional to the square of the amplitude of the wave.

This energy-amplitude relationship is sometimes expressed in the following manner. This means that a doubling of the amplitude of a wave is indicative of a quadrupling of the energy transported by the wave.

A tripling of the amplitude of a wave is indicative of a nine-fold increase in the amount of energy transported by the wave. And a quadrupling of the amplitude of a wave is indicative of a fold increase in the amount of energy transported by the wave.

The table at the right further expresses this energy-amplitude relationship. Observe that whenever the amplitude increased by a given factor, the energy value is increased by the same factor squared. For example, changing the amplitude from 1 unit to 2 units represents a 2-fold increase in the amplitude and is accompanied by a 4-fold 22 increase in the energy; thus 2 units of energy becomes 4 times bigger - 8 units.

As another example, changing the amplitude from 1 unit to 4 units represents a 4-fold increase in the amplitude and is accompanied by a fold 42 increase in the energy; thus 2 units of energy becomes 16 times bigger - 32 units. Earthquakes and other geologic disturbances sometimes result in the formation of seismic waves. Seismic waves are waves of energy that are transported through the earth and over its surface.

Earthquakes are given a Richter scale rating that indicates how intense the earthquake is. Use the Earthquake Energy widget below to explore the relationship between the Richter scale magnitude and the amount of energy transmitted by seismic waves.

Check Your Understanding 1. Mac and Tosh stand 8 meters apart and demonstrate the motion of a transverse wave on a snakey. The wave can be described as having a vertical distance of 32 cm from a trough to a crest, a frequency of 2.