A proton and an alpha particle are released from rest when they are 0.225 nm apart. The alpha pa

step 1 So in this question we have two different particles. One is an alpha particle, or that means a h

A proton and an alpha particle are released from rest when...

Key Concepts

Two-Body Problem in Electrostatics

The two-body problem in electrostatics involves analyzing the motion of both interacting particles simultaneously. In such problems, conservation laws (like momentum and energy) are used along with individual equations of motion to determine quantities such as maximum speed and acceleration. The different masses and charges affect how each particle responds to the electrostatic force in a way that respects both conservation of momentum and energy.

Dynamics of Charged Particles

The dynamics of charged particles involves understanding how forces act on the particles and how their masses and charges determine their accelerations and motions. The acceleration at any point is given by Newton’s second law, where the force from the electric field is divided by the particle’s mass. This is particularly important when analyzing situations where particles have different masses and charges.

Dependence of Acceleration on Distance

In systems where forces obey an inverse-square law, such as electrostatic forces, acceleration is greatest when the particles are closest together. This means that the maximum acceleration occurs at the initial moment when the distance between the charges is minimal. As the particles move apart, the force and consequently the acceleration decrease.

Conservation of Energy

The conservation of energy principle explains that as the charged particles repel or attract each other, the total energy of the system (the sum of kinetic and potential energies) remains constant. In problems involving electric forces, the initial electrostatic potential energy is gradually converted into kinetic energy as the particles accelerate. This principle is crucial for determining the maximum speed of the particles from the initial potential energy available.

Coulomb's Law

Coulomb's Law states that the electrostatic force between two point charges is proportional to the product of the magnitudes of the charges and inversely proportional to the square of the distance between them. This principle is used to determine the force acting on charged particles, which is fundamental in calculating both accelerations and energy exchanges in systems of charged particles.

Transcript

00:01 So in this question we have two different particles.

00:03 One is an alpha particle, or that means a helium atom, and one is a proton.

00:10 So the proton has a charge of positive e, and it has a mass of m.

00:16 The alpha particle, which is a helenucl, has four times the mass, and two times the charge.

00:24 They are released from rest, but then they started traveling in opposite directions.

00:29 We want to find the maximum speed and maximum acceleration for each of these.

00:34 And we want to know when these maximum values occur.

00:38 So the easiest thing to find first is the maximum acceleration.

00:43 Why? because when these two particles are first released, they will have a force on each other, and it's this proportion of force and makes them start to move.

00:52 But this force f equals kq1, q2 over our square.

00:59 This force decreases the further apart from each other.

01:04 So this force is the largest when they're closest, beginning at the very beginning.

01:10 So they will also achieve their maximum acceleration from this force because a equals over m, kq1q2 over mr squared.

01:23 So because the force is largest, when they were just released, their acceleration will be largest when they're just released.

01:30 So the acceleration of an alpha particle is k times is 2 k squared because it's e times 2 e over 4 times n r squared, where m is the mass of a single proton.

01:51 So this is just 2.

01:58 Yeah, so okay, so it's 2 times the mass is 1.

02:07 7 times 10 to the negative 27 kilogram times r is 0 .25 times 10 to the negative 9 meters and there is a square and the numerator is 8 times 10 to the 9s newton meter square per quillum square times 1 .6.

02:40 Times 10 to the negative 19 cullen gain square.

02:45 Okay, and the acceleration is 1 .36 times 10 to the 18 meter per second square.

02:57 So the acceleration for proton is, again, it's 2ke square, it's also e times 2e, and now we have a normal mr square.

03:09 So the numerator is the same times 1 % oh, i don't forget to add the units.

03:20 T square times 2 over 1 .67 times 102 and negative 27 times 0 .225 times 10 to the negative 9 square.

03:38 Okay, so this end up being 5 .5, 5 .4.

03:49 Times 10 to the 18 meter per second square.

03:53 So this you can see is four times larger than the previous one and that actually satisfies the conservation of momentum.

04:04 Yeah, so the conservation of momentum we will see is what we need for finding the speed.

04:16 Like we can already see it from the acceleration because this momentum has to add up to zero because it was initially at rest.

04:25 The speed of the m particle must be four times larger, so the speed of the 4m particle, because m times v of proton plus 4m times v of the alpha particle is 0...

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