A point charge Q= 4.60 uC is held fixed at the origin. A second point charge q=1.20 uC with mass of 2.80 * 10^-4 is placed on the x-axis 0.250 m away from the origin. (b) The second point charge is released from rest. What is its speed withn its distance from the origin is (i) .500 m (ii) 5.00 m (iii) 50.0 m

I am not sure how to get the speed from the different distances

A quick way to do this is to use conservation of energy. The potential energy of the moving charge in the E-field of the fixed charge is:

P.E. = k Q q/R
where k = 8.99*10^9 N m^2/C^2
The kinetic energy of the particle is
(1/2) m V^2

The initial energy, which is all potential, is the total energy, which is constant. Therefore
kQq/(0.25) = kQq/r + (1/2) m V^2

V = sqrt [(2/m)kQq (4 - 1/r)]

Solve for V at each or the r values in your question 1/2, 5 and 50 m.

To find the speed of the second point charge at different distances from the origin, we need to determine the electrostatic potential energy and then use the principle of conservation of energy.

The electrostatic potential energy (PE) between two point charges can be calculated using the formula:

PE = (k * Q * q) / r

where:
k is the electrostatic constant (8.99 × 10^9 N m^2/C^2)
Q is the charge of the fixed point charge (4.60 μC = 4.60 × 10^-6 C)
q is the charge of the second point charge (1.20 μC = 1.20 × 10^-6 C)
r is the distance between the charges

Let's calculate the electrostatic potential energy at the initial distance of 0.250 m:

PE_initial = (k * Q * q) / r_initial
= (8.99 × 10^9 N m^2/C^2) * (4.60 × 10^-6 C) * (1.20 × 10^-6 C) / 0.250 m

Now, using the conservation of energy, we can equate the potential energy at the initial distance to the kinetic energy at any other distance. The kinetic energy (KE) is given by:

KE = (1/2) * m * v^2

where:
m is the mass of the second point charge (2.80 × 10^-4 kg)
v is the velocity of the second point charge

We can equate the potential energy and kinetic energy to solve for the velocity (v).

PE_initial = KE
(8.99 × 10^9 N m^2/C^2) * (4.60 × 10^-6 C) * (1.20 × 10^-6 C) / 0.250 m = (1/2) * (2.80 × 10^-4 kg) * v^2

Now, let's calculate the velocity (v) at different distances:

(i) Distance = 0.500 m:

PE_final = (k * Q * q) / r_final
= (8.99 × 10^9 N m^2/C^2) * (4.60 × 10^-6 C) * (1.20 × 10^-6 C) / 0.500 m

Using the conservation of energy equation:

(8.99 × 10^9 N m^2/C^2) * (4.60 × 10^-6 C) * (1.20 × 10^-6 C) / 0.250 m = (1/2) * (2.80 × 10^-4 kg) * v^2

Solve for v.

(ii) Distance = 5.00 m:
Repeat the same calculation above using r_final = 5.00 m.

(iii) Distance = 50.0 m:
Repeat the same calculation above using r_final = 50.0 m.

To find the speed of the second point charge at various distances from the origin, we can use the principle of conservation of mechanical energy.

The mechanical energy of the system is conserved when there is no external force acting on it. In this case, as the point charge is released from rest and moves along the x-axis, the only force acting on it is the electrostatic force due to the fixed charge at the origin. Since the electrostatic force is conservative, we can apply the principle of conservation of mechanical energy.

The mechanical energy of the system consists of two components: potential energy (due to the interaction between the charges) and kinetic energy (due to the motion of the second point charge).

The potential energy of the system is given by the formula:
PE = k * (Q * q) / r

Where k is the electrostatic constant (9 * 10^9 Nm^2/C^2), Q is the charge at the origin (4.60 uC = 4.60 * 10^-6 C), q is the charge at the distance (1.20 uC = 1.20 * 10^-6 C), and r is the distance between the charges.

The kinetic energy of the system is given by the formula:
KE = (1/2) * m * v^2

Where m is the mass of the second point charge (2.80 * 10^-4 kg) and v is its speed.

Since the mechanical energy is conserved, we can equate the initial and final mechanical energies:

PE_initial + KE_initial = PE_final + KE_final

Since the second point charge is at rest initially, its initial kinetic energy is zero. Therefore, the equation simplifies to:

PE_initial = PE_final + KE_final

Solving for KE_final, we get:

KE_final = PE_initial - PE_final

Now we can calculate the potential energy at the initial and final distances.

For (i) a distance of 0.500 m:
PE_initial = k * (Q * q) / r_initial
PE_final = k * (Q * q) / r_final

Substituting the values into the formula, we get:
KE_final = PE_initial - PE_final

For (ii) a distance of 5.00 m:
PE_initial = k * (Q * q) / r_initial
PE_final = k * (Q * q) / r_final

Substituting the values into the formula, we get:
KE_final = PE_initial - PE_final

And similarly for (iii) a distance of 50.0 m.

Once we have the value of KE_final, we can solve for the speed (v) using the formula:
v = √(2 * KE_final / m)

With this approach, you can calculate the speed of the second point charge at different distances from the origin.