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?
k =9•10⁹ N•m²/C²,
q=4.8•10⁻⁶ C
m=5.5•10⁻³ kg
r=0.88 m
PE₁-PE₂=2KE
-kq²/r +3kq²/r =2mv²/2
2kq²/r =2mv²/2
kq²/r =mv²/2
v=sqrt( kq²/rm) =
=sqrt{9•10⁹•(4.8•10⁻⁶)²/0.88•5.5•10⁻³} =
=6.55 m/s
To find the speed of each particle when the separation between them is one-third its initial value, we can use the principle of conservation of energy. The initial potential energy of the system is equal to the final kinetic energy of the two particles.
Let's break down the steps to find the answer:
Step 1: Find the initial potential energy of the system.
The potential energy between two charges can be calculated using the formula:
Potential Energy (PE) = (k * |q1 * q2|) / r
where k is the electrostatic constant (8.99 x 10^9 Nm^2/C^2), q1 and q2 are the charges (+4.8 x 10^-6 C and -4.8 x 10^-6 C, respectively), and r is the separation between the charges (0.88 m).
So,
PE_initial = (8.99 x 10^9 Nm^2/C^2) * |(+4.8 x 10^-6 C) * (-4.8 x 10^-6 C)| / 0.88 m
Step 2: Find the final potential energy of the system.
When the separation between the charges is reduced to one-third its initial value, the separation becomes (1/3) * 0.88 m. Using this new separation, we can find the final potential energy.
PE_final = (8.99 x 10^9 Nm^2/C^2) * |(+4.8 x 10^-6 C) * (-4.8 x 10^-6 C)| / [(1/3) * 0.88 m]
Step 3: Apply the principle of conservation of energy.
According to the principle of conservation of energy, the initial potential energy (PE_initial) of the system is equal to the final kinetic energy (KE) of the particles.
PE_initial = KE_particle1 + KE_particle2
The kinetic energy of each particle can be calculated using the formula:
KE = (1/2) * m * v^2
where m is the mass of the particle (5.5 x 10^-3 kg) and v is the velocity of the particle.
Substituting the formula for kinetic energy into the conservation of energy equation, we get:
PE_final = (1/2) * m_particle1 * v_particle1^2 + (1/2) * m_particle2 * v_particle2^2
Step 4: Solve for velocities.
Now we have two unknowns, v_particle1 and v_particle2. We can solve these equations simultaneously to find their values.
v_particle1 = sqrt((2 * (PE_final - (1/2) * m_particle2 * v_particle2^2)) / m_particle1)
v_particle2 = sqrt((2 * (PE_final - (1/2) * m_particle1 * v_particle1^2)) / m_particle2)
Substituting the values we have:
v_particle1 = sqrt((2 * (PE_final - (1/2) * (5.5 x 10^-3 kg) * v_particle2^2)) / (5.5 x 10^-3 kg))
v_particle2 = sqrt((2 * (PE_final - (1/2) * (5.5 x 10^-3 kg) * v_particle1^2)) / (5.5 x 10^-3 kg))
Simplify these equations further and solve for v_particle1 and v_particle2.
Calculating these values will give us the speeds at which each particle is moving when the separation between them is one-third of its initial value.