Question

You are strapped into a rear-facing seat at the middle of a long bus accelerating from rest at about \(10 \mathrm{~m} / \mathrm{s}^{2}\) (a rather violent acceleration for a bus). As the back of the bus passes a warning sign alongside the street, a red light of precisely 650 nm wavelength on the sign tums on. Do you see this precise 650 nm wavelength? Does your friend sitting at the front of the bus see the wavelength you see? How could the same observations be produced with the bus and sign stationary?

Short answer

Neither you nor your friend would see the exact 650nm wavelength due to the Doppler effect caused by the movement of the bus. The person in the middle of the bus would observe a slightly reduced wavelength, and the person at the front would observe an even lower wavelength light. To simulate the same observations with a stationary bus and sign, light sources of lower wavelengths are needed to replace the 650nm light.

Step-by-step solution

Analyze the Problem's Conditions

The bus is being accelerated, which means it is moving with a velocity \(v\). This will cause a Doppler shift in the frequency of light observed, which will be different for the observer in the middle of the bus and that at the front due to their relative velocities.

Calculate Doppler Shift for Observer in the Middle

An observer inside the bus who is moving at velocity \(v\) towards the source of light will observe a blue shift (decrease in wavelength). The Doppler shift can be expressed using the formula \( \Delta\lambda = \lambda - \lambda_0 = \lambda_0(\sqrt{(1+\beta)/(1-\beta)} - 1) \) where \( \beta = v/c \) (c is the speed of light), \( \lambda \) is the observed wavelength and \( \lambda_0 \) is the source wavelength. Therefore, the observer will not see the exact wavelength of 650 nm, i.e., their observed wavelength will be less than 650 nm due to blue shift.

Calculate Doppler Shift for Observer at the Front

The observer sitting at the front of the bus will see the light after it has passed the observer sitting in the middle, when the bus has a higher velocity. Hence, the frequency of the light observed by this observer will be even more blue-shifted, i.e., their observed wavelength will be even lesser than that observed by the person sitting in the middle of the bus.

Simulate Same Observations with Stationary Bus and Sign

In order to simulate the same observations with a stationary bus and sign, the 650 nm light would need to be replaced by a higher frequency or lower wavelength light. This is due to the fact that without the relative motion, there would be no Doppler shift causing the blue shift.

Key concepts

Blue Shift

When you are moving towards a light source, something interesting happens. The light seems to change color. This change is called a blue shift. It gets its name because the light's wavelength appears shorter. Shorter wavelengths are toward the blue end of the visible spectrum.

Blue shift happens because of the Doppler Effect. The Doppler Effect is not just about sound waves, like when an ambulance passes you with its siren blaring. It also applies to light. As you approach a light source, the waves get compressed, resulting in a blue shift.

• Shortening of the wavelength
• Increase in frequency
• No actual color change; it's all about perception

From our exercise example, the observer in the bus moving towards a red light sees a shift towards the blue end of the spectrum, not actually turning colors but appearing with a different wavelength.

Wavelength

Wavelength is a fundamental concept when talking about waves, whether you are dealing with sound or light. It describes the distance between consecutive peaks of a wave.

In the exercise, the light has a wavelength of 650 nm, which is characteristic of red light. When a wavelength shortens, it means the peaks of the waves are getting closer together. This can be caused by motion, as explained in the blue shift concept.

• Measured in units of distance, such as nanometers (nm)
• Key property in identifying types of light
• Changes perceived color when observing a moving source

Understanding wavelength helps to understand why movement affects color perception during the Doppler effect.

Frequency

Frequency is the number of wave peaks that pass a point in one second. Related to wavelength, frequency tells us how often the waves are hitting a certain point.

There is an inverse relationship between frequency and wavelength: when one goes up, the other goes down. If a wave's frequency increases, its wavelength decreases, and vice versa. In the exercise, as the bus accelerates towards the light source, the frequency of the observed light increases.

• Measured in Hertz (Hz), which means "per second"
• Higher frequency means more energy carried by the wave
• Increased frequency corresponds to a blue shift

This change in frequency explains why the observer sees a different color when they move closer to the light source, not because the light source changes, but because their motion changes the wave observations.

Velocity

Velocity refers to the speed and direction of an object’s movement. It plays a crucial role in the Doppler Effect acting on waves.

In the discussed exercise, the velocity of the bus influences how the light appears to different observers. The higher the velocity towards the light source, the more significant the blue shift observed.

• Determines how fast an observer is moving towards or away from a light source
• The cause of wavelength change in the Doppler Effect
• Expressed in units like meters per second (m/s)

Understanding velocity's role helps us to calculate how significant the change in observed wavelength would be, just like in the case with our bus and the red light.