Two particles each have a mass of 5.5 10-3 kg. One has a charge of +4.8 10-6 C, and the other has a charge of -4.8 10-6 C. They are initially held at rest at a distance of 0.88 m apart. Both are then released and accelerate toward each other. How fast is each particle moving when the separation between them is one-third its initial value?
To solve this problem, we can use the principles of electrostatics and conservation of energy. Let's break it down step by step:
1. Determine the initial potential energy:
The initial potential energy of the system is given by the equation:
PE = k * (q1 * q2) / r
where:
k is the Coulomb's constant (8.99 × 10^9 N·m^2/C^2)
q1 and q2 are the charges of the particles (+4.8 × 10^-6 C and -4.8 × 10^-6 C, respectively)
r is the initial separation distance (0.88 m)
Plugging in the values, we get:
PE_initial = (8.99 × 10^9 N·m^2/C^2) * [(+4.8 × 10^-6 C) * (-4.8 × 10^-6 C)] / (0.88 m)
2. Determine the final potential energy:
When the separation between the particles is one-third its initial value, the new separation distance is (1/3) * 0.88 m = 0.293 m.
Using the same equation as above, but with the new separation distance, we can calculate the final potential energy (PE_final).
3. Apply the principle of conservation of energy:
According to the conservation of energy, the initial potential energy is converted into the final kinetic energy of the system when the particles start moving. Therefore, we equate the initial potential energy to the sum of the final kinetic energies of both particles.
PE_initial = KE_particle1 + KE_particle2
The kinetic energy of a particle can be calculated using the equation:
KE = (1/2) * mv^2
where m is the mass of the particle (5.5 × 10^-3 kg) and v is its velocity.
4. Calculate the final velocities:
Let v1 and v2 be the final velocities of particle 1 and particle 2, respectively.
Solving the equation from step 3 for v1 and v2, we have:
(1/2) * m * v1^2 + (1/2) * m * v2^2 = PE_final
Since both particles have the same mass, we can simplify this equation to:
m * (v1^2 + v2^2) = 2 * PE_final
Now, we can rearrange the equation to solve for v1 and v2:
v1^2 + v2^2 = (2 * PE_final) / m
Taking the square root of both sides, we get:
v1 = sqrt((2 * PE_final) / m)
v2 = sqrt((2 * PE_final) / m)
5. Substitute the values and calculate the final velocities:
Substitute the value of PE_final obtained in step 2, along with the mass of the particles, into the formulas for v1 and v2 to calculate their velocities.
v1 = sqrt((2 * PE_final) / (5.5 × 10^-3 kg))
v2 = sqrt((2 * PE_final) / (5.5 × 10^-3 kg))
Once you calculate these values, you will have the final velocities of both particles.