Calculating Speed of Charged Particles in Electric Field: An Example"

[StudentofPhysics]

Two particles each have a mass of 5.3 10-3 kg. One has a charge of +5.0 10-6 C, and the other has a charge of -5.0 10-6 C. They are initially held at rest at a distance of 0.75 m apart. Both are then released and accelerate toward each other. How fast is each particle moving when the separation between them is one-half its initial value?


ok, I know:
q1 = +5.0 x 10^-6 C
q2 = -5.0 x 10^-6 C
r = 0.75m
m = 5.3 x 10^-3 kg

Since they are opposite charges they will atract each other.

I can't seem to figure out where to begin on what formula to use to discover the speed without any time.

 

[fargoth]

the fastest way would be transfering the potential energy difference to kinetic energy.

$$\Delta U=\frac{mv^2}{2}+\frac{mv^2}{2}$$

if they didn't have the same mass you had to use the momentum conservation eq. aswell.

 

[kyle8921]

I would first start by identifying the key variables and principles involved in this scenario. We have two charged particles with known masses and charges, initially held at rest and then released to accelerate towards each other in an electric field. The distance between them is also given, and we are asked to find the speed of each particle when the separation between them is halved.

To solve this problem, we can use the principles of electrostatics and Newton's laws of motion. The electric force between two charged particles is given by Coulomb's law, which states that the force is directly proportional to the product of the charges and inversely proportional to the square of the distance between them. This can be represented mathematically as:

F = k * (q1 * q2) / r^2

where k is the Coulomb's constant (8.99 x 10^9 N*m^2/C^2), q1 and q2 are the charges of the particles, and r is the distance between them.

Since the particles are accelerating towards each other, we can also use Newton's second law, which states that the net force acting on an object is equal to its mass multiplied by its acceleration. This can be represented mathematically as:

F = m * a

where m is the mass of the particle and a is its acceleration.

Now, we can set these two equations equal to each other and solve for the acceleration:

k * (q1 * q2) / r^2 = m * a

a = (k * (q1 * q2) / (m * r^2)

Plugging in the given values, we get:

a = (8.99 x 10^9 N*m^2/C^2) * ((5.0 x 10^-6 C) * (-5.0 x 10^-6 C)) / ((5.3 x 10^-3 kg) * (0.75 m)^2)

a = -3.02 x 10^11 m/s^2

This negative acceleration indicates that the particles are accelerating towards each other, as expected.

Now, to find the speed of each particle when the separation between them is halved, we can use the equation for constant acceleration:

v^2 = u^2 + 2 * a * s

where v is the final velocity, u is the initial velocity (which is 0 in this