Answer :
Sure, let's analyze the problem step by step:
1. Understanding the Variables:
- Total energy and momentum of a collision: This is the sum of all energy and momentum resulting from the collision event. Let’s denote it by [tex]\( E_{\text{total}} \)[/tex].
- Energy and momentum contained in the particles that collided: This represents the energy and momentum that can be accounted for by the particles observed after the collision. Let’s denote it by [tex]\( E_{\text{particles}} \)[/tex].
- Dark matter: This is the difference between the total energy and momentum and the energy and momentum accounted for by the observed particles, representing the "missing" energy and momentum not accounted for by visible particles. We’ll denote this as [tex]\( E_{\text{dark}} \)[/tex].
2. Formulating the Equation:
- According to the numerical values given:
- [tex]\( E_{\text{total}} = 100 \)[/tex]
- [tex]\( E_{\text{particles}} = 70 \)[/tex]
- [tex]\( E_{\text{dark}} = \)[/tex] (Total energy) - (Particles' energy) = [tex]\( 100 - 70 = 30 \)[/tex]
3. Choosing the Correct Equation:
- Let's now assess the provided options:
a. Dark matter + total energy and momentum of a collision = 100
- This equation suggests that the sum of dark matter and the total energy and momentum of the collision is always 100. In our context, that would imply [tex]\( E_{\text{dark}} + E_{\text{total}} = 100 \)[/tex], which doesn't make sense because [tex]\( E_{\text{total}} \)[/tex] is already defined as 100.
b. Total energy + total momentum = dark matter
- This option incorrectly states the relationship between total energy, total momentum, and dark matter.
c. Total energy and momentum of a collision - energy and momentum contained in the particles that collided = dark matter
- Rearranging this, it aligns exactly with our earlier statement [tex]\( E_{\text{dark}} = E_{\text{total}} - E_{\text{particles}} \)[/tex].
d. Dark matter - energy missing from a collision = energy contained in the particles that collided
- This equation introduces a new variable "energy missing from a collision", which can be confused with dark matter itself. It's not a suitable representation based on the provided context.
4. Conclusion:
- The correct choice is:
c. Total energy and momentum of a collision - energy and momentum contained in the particles that collided = dark matter.
This accurately reflects the relationship between the total energy and momentum of a collision, the observed energy and momentum of the particles, and the dark matter detected.
1. Understanding the Variables:
- Total energy and momentum of a collision: This is the sum of all energy and momentum resulting from the collision event. Let’s denote it by [tex]\( E_{\text{total}} \)[/tex].
- Energy and momentum contained in the particles that collided: This represents the energy and momentum that can be accounted for by the particles observed after the collision. Let’s denote it by [tex]\( E_{\text{particles}} \)[/tex].
- Dark matter: This is the difference between the total energy and momentum and the energy and momentum accounted for by the observed particles, representing the "missing" energy and momentum not accounted for by visible particles. We’ll denote this as [tex]\( E_{\text{dark}} \)[/tex].
2. Formulating the Equation:
- According to the numerical values given:
- [tex]\( E_{\text{total}} = 100 \)[/tex]
- [tex]\( E_{\text{particles}} = 70 \)[/tex]
- [tex]\( E_{\text{dark}} = \)[/tex] (Total energy) - (Particles' energy) = [tex]\( 100 - 70 = 30 \)[/tex]
3. Choosing the Correct Equation:
- Let's now assess the provided options:
a. Dark matter + total energy and momentum of a collision = 100
- This equation suggests that the sum of dark matter and the total energy and momentum of the collision is always 100. In our context, that would imply [tex]\( E_{\text{dark}} + E_{\text{total}} = 100 \)[/tex], which doesn't make sense because [tex]\( E_{\text{total}} \)[/tex] is already defined as 100.
b. Total energy + total momentum = dark matter
- This option incorrectly states the relationship between total energy, total momentum, and dark matter.
c. Total energy and momentum of a collision - energy and momentum contained in the particles that collided = dark matter
- Rearranging this, it aligns exactly with our earlier statement [tex]\( E_{\text{dark}} = E_{\text{total}} - E_{\text{particles}} \)[/tex].
d. Dark matter - energy missing from a collision = energy contained in the particles that collided
- This equation introduces a new variable "energy missing from a collision", which can be confused with dark matter itself. It's not a suitable representation based on the provided context.
4. Conclusion:
- The correct choice is:
c. Total energy and momentum of a collision - energy and momentum contained in the particles that collided = dark matter.
This accurately reflects the relationship between the total energy and momentum of a collision, the observed energy and momentum of the particles, and the dark matter detected.
