Question

An object moving to the right at \(0.8 c\) issouck head-on by a photon of wavelength \(\lambda\) moving to the left. The An object moving to the right at \(0.8 c\) is souck head-on by a photon of wavelength \(\lambda\) moving to the left. The object absorbs the photon (i.e., the photon disappears) and is afterward moving to the right at \(0.6 c\). (a) Determine the ratio of the object's mass after the collision to its mass before the collision. (Note: The object is not a "fundamental particle." and its mass is therefore subject lo change.) (b) Does kinetic energy inc rease or decrease?

Short answer

a) The requested ratio is determined by substituting the given values into the formula obtained in Step 2. The exact result will depend on the wavelength of the incoming photon, which is not specified. b) The kinetic energy decreases as a result of the collision.

Step-by-step solution

Analyze the problem and set up the equations

The object absorbs a photon and changes its speed, implying that both its mass and momentum may change. By conservation of linear momentum, the total momentum before the collision (the sum of the object and photon momenta) must equal the total momentum after the collision. Before the collision, the object's momentum is \(m_0 \cdot v_0\) (where \(m_0\) is the original mass of the object and \(v_0\) is its original speed) and the photon's momentum is \(\frac{h}{\lambda}\) (where \(h\) is Planck’s constant and \(\lambda\) is the wavelength of the photon). After the collision, the object's momentum is \(m_1 \cdot v_1\) (where \(m_1\) is the mass of the object after collision and \(v_1\) is its speed after collision). Thus the conservation equation becomes: \(m_0 \cdot v_0 - \frac{h}{\lambda} = m_1 \cdot v_1\).

Calculate the mass of the object after collision

By algebraic manipulation, the ratio of the object's mass after the collision to its mass before the collision (\(m_1/m_0\)) can be determined: \(m_1/m_0 = \frac{v_0 - (h/(\lambda m_0))}{v_1}\). Replacing \(v_0\), \(v_1\), \(h\), and \(\lambda\) with their given values will provide the solution.

Determine the change in kinetic energy

Since the object's speed decreases after the collision (from \(0.8c\) to \(0.6c\)), it is expected that its kinetic energy decreases as well. However, to confirm this, check the difference between the initial and final kinetic energies using the relativistic kinetic energy formula: \(K.E = mc^2 – m_0c^2\), where \(m\) is the relativistic mass, \(c\) is the speed of light, and \(m_0\) is the rest mass.

Key concepts

Relativistic Kinetic Energy

When objects travel at speeds close to the speed of light, classical physics doesn't quite cut it. This is where relativistic kinetic energy comes in handy. Unlike classical kinetic energy, which depends only on mass and velocity, relativistic kinetic energy also considers the effects of special relativity. For a moving object, the formula is \( K.E = mc^2 – m_0c^2 \), where:

• \( m \) is the relativistic mass of the object.
• \( m_0 \) is the rest mass (mass when the object is not moving).
• \( c \) is the speed of light.

In the given problem, since the object's speed reduces from \(0.8c\) to \(0.6c\), we expect the kinetic energy to decrease.
However, using the formula confirms this decrease as we calculate kinetic energy before and after the collision. The reduction in speed limits the energy the object can have due to that dependance and shows the significance of incorporating relativity for a complete picture.

Photon Momentum

A photon, though massless, can carry momentum. This might seem odd initially, but according to quantum mechanics, the momentum of a photon is derived from its wave properties. The formula is \( p = \frac{h}{\lambda} \), where:

• \( p \) is the momentum of the photon.
• \( h \) is Planck's constant (approximately \( 6.626 \times 10^{-34} \) Js).
• \( \lambda \) is the wavelength of the photon.

In the scenario provided, a photon strikes the object, transferring its momentum.
This affects the overall momentum conservation in the collision. Though photons lack mass, their interactions in collisions still impact the speeds and, in cases like this, the mass of objects they collide with. Understanding photon momentum helps explain why the object changed velocity after absorption.

Mass-Energy Equivalence

Mass-energy equivalence is a cornerstone of Einstein's theory of relativity. It suggests that mass and energy are two forms of the same thing, described by the equation \( E = mc^2 \). This means:

• Energy can be thought of as a 'hidden' form of mass.
• As energy increases, so does the effective mass.
• The conversion between mass and energy is direct and determined by the constant \( c^2 \).

In the problem, the object absorbs a photon, which contributes energy. This energy might cause the object's mass to increase, even if it appears momentarily.
Such principles allow us to understand how mass can change in non-fundamental particles, as noted in the exercise, since the energy carried by the absorbed photon effectively turns into an increase in mass of the object, observed as \( m_1 \) after the collision.