Two protons and two positrons form a square as shown in the figure. All the particles are initially at rest. Due to Coulomb's repulsion, the particles fly off to infinity. Determine the ratio vp/ve where vp and ve are the final velocities of the protons and positrons, respectively. Note that the positron, being electron's antiparticle, has the same mass as the electron and opposite charge (q=+1.6×10^−19C).Details and assumptions
mp/me=1836
Hint: since the protons are so much heavier, you can do this without having to solve momentum conservation.
To determine the ratio vp/ve, we can take advantage of the fact that the protons are much heavier than the positrons. This allows us to neglect momentum conservation and focus solely on the electrostatic repulsion between the particles.
Let's analyze the forces acting on each particle. Each proton experiences a repulsive force from the other proton and both positrons, while each positron experiences a repulsive force from the other positron and both protons.
Since all the particles are initially at rest, the electrostatic repulsion between them is the only force acting on them, and it causes them to fly off to infinity.
The magnitude of the electrostatic force between two charged particles is given by Coulomb's Law:
F = k * (q1 * q2) / r^2
Where:
F is the magnitude of the force
k is the electrostatic constant (k = 8.988 × 10^9 N m^2/C^2)
q1 and q2 are the charges of the two particles involved
r is the distance between the particles
In this case, the charges of the particles are:
q(proton) = +1.6 × 10^-19 C
q(positron) = +1.6 × 10^-19 C
Since the particles form a square, the distance between each pair of particles is the same. Let's denote this distance as d.
Now, let's analyze the forces acting on each particle:
Each proton experiences a repulsive force from two positrons and one other proton:
F(proton) = 2 * F(positron) + F(proton)
Each positron experiences a repulsive force from two protons and one other positron:
F(positron) = 2 * F(proton) + F(positron)
Given that the mass of the positron is the same as the electron (me):
m(proton) = 1836 * m(electron)
Using the equation F = m * a, we can relate the forces to the resulting accelerations:
For protons:
2 * F(positron) + F(proton) = m(proton) * a(proton)
For positrons:
2 * F(proton) + F(positron) = m(electron) * a(positron)
Since the particles fly off to infinity, their final velocities are both equal to infinity. This means that their final accelerations are also infinite.
Now, we have two equations, one for protons and one for positrons:
2 * F(positron) + F(proton) = m(proton) * a(proton)
2 * F(proton) + F(positron) = m(electron) * a(positron)
We can plug in the values of the charges and the electrostatic force expressions:
2 * (k * (q(positron))^2 / d^2) + (k * (q(proton))^2 / d^2) = m(proton) * a(proton)
2 * (k * (q(proton))^2 / d^2) + (k * (q(positron))^2 / d^2) = m(electron) * a(positron)
Simplifying:
2 * (1.6 × 10^-19)^2 / d^2 + (1.6 × 10^-19)^2 / d^2 = 1836 * (1.6 × 10^-19) * a(proton)
2 * (1.6 × 10^-19)^2 / d^2 + (1.6 × 10^-19)^2 / d^2 = (1.6 × 10^-19) * a(positron)
Note that we can cancel out the (1.6 × 10^-19)^2 terms on both sides.
Now, we have two equations in terms of the final accelerations:
2 / d^2 + 1 / d^2 = 1836 * a(proton)
2 / d^2 + 1 / d^2 = a(positron)
Simplifying further:
3 / d^2 = 1836 * a(proton)
3 / d^2 = a(positron)
Since the accelerations are equal to infinity, we can simplify the equations to:
a(proton) = infinity
a(positron) = infinity
Therefore, the ratio vp/ve can be expressed as:
vp / ve = infinity / infinity
The ratio of the final velocities of the protons to the positrons cannot be determined based solely on the given information and assumptions.