One particle has a mass of 3.00*10-3 kg and a charge of +7.60 µC. A second particle has a mass of 6.00*10-3 kg and the same charge. The two particles are initially held in place and then released. The particles fly apart, and when the separation between them is 0.100 m, the speed of the 3.00*10-3 kg particle is 140 m/s. Find the initial separation between the particles.
You can so,ve this problem by using conservation of energy.
First you find the speed of the aother particle. You use the fact that momentum is conserved. It was zero when the particles were released so it's still zero.
The fact that the total mometum is zero leads to the conclusion that second particle is moving at half the speed of the first particle (and in the opposite direction), because it's mass is twice that of the first particle.
You now know the speeds of both particles and from that you can calculate the kinetic energy. From the distance of the two particles you calculate the potential energy:
q1*q2/(4 pi epsilon_0 r)
The total energy is the sum of the potential and kinetic energy. When the particles were just released they both had a velocity of zero. So at that time the total energy was the potential energy they had. But since energy is conserved that potential energy was equal to the total energy you just computed. This means that the distance r_int is such that:
q1*q2/(4 pi epsilon_0 r_int) = total energy
To find the initial separation between the particles, we can follow these steps:
Step 1: Find the speed of the second particle
Since momentum is conserved in the system, the total momentum of the particles before and after they fly apart is zero.
The initial momentum is zero because the particles are initially at rest.
Since the masses of the two particles are in a 2:1 ratio, the second particle must have half the speed of the first particle (but in the opposite direction).
Let the speed of the first particle be v1. Therefore, the speed of the second particle (v2) is -0.5v1.
Step 2: Calculate the kinetic energy of both particles
The kinetic energy (KE) of a particle is given by the formula:
KE = (1/2) * mass * velocity^2
For the first particle (m1 = 3.00*10^-3 kg), the kinetic energy is:
KE1 = (1/2) * (3.00*10^-3 kg) * (140 m/s)^2
For the second particle (m2 = 6.00*10^-3 kg), the kinetic energy is:
KE2 = (1/2) * (6.00*10^-3 kg) * (-0.5v1)^2 = (3/8) * (6.00*10^-3 kg) * v1^2
Step 3: Calculate the potential energy between the particles
The potential energy (PE) between two charged particles is given by Coulomb's Law:
PE = (k * q1 * q2) / r
Where k is the electrostatic constant, q1 and q2 are the charges of the particles, and r is the separation between them.
Using the values given in the problem, and assuming k = 9.0 * 10^9 Nm^2/C^2, the potential energy is:
PE = (9.0 * 10^9 Nm^2/C^2) * (7.60 µC)^2 / (0.100 m)
Step 4: Find the total energy of the system
The total energy (E) of the system is the sum of the kinetic and potential energy:
E = KE1 + KE2 + PE
Step 5: Equate the total energy to the potential energy at the initial separation
At the initial separation, when the particles are just released, their kinetic energy is zero.
Therefore, the total energy at that moment is equal to the potential energy:
E = PE = (9.0 * 10^9 Nm^2/C^2) * (7.60 µC)^2 / r_int
Step 6: Solve for the initial separation (r_int)
Equating the expressions for E and PE, we can solve for r_int:
(9.0 * 10^9 Nm^2/C^2) * (7.60 µC)^2 / r_int = KE1 + KE2 + PE
Solve the equation for r_int, which will give you the initial separation between the particles.