A proton accelerates from rest in a uniform electric field of 640 N / C . At some later time, it

A proton accelerates from rest in a uniform electric field...

A proton accelerates from rest in a uniform electric field of 640 $\mathrm{N} / \mathrm{C}$ . At some later time, its speed is $1.20 \times 10^{6} \mathrm{m} / \mathrm{s}$ (nonrelativistic, because $v$ is much less than the speed of light). (a) Find the acceleration of the proton. (b) How long does it take the proton to reach this speed? (c) How far has it moved in this time? (d) What is its kinetic energy at this time?

A proton accelerates from rest in a uniform electric field of 640 $\mathrm{N} / \mathrm{C}$ . At some later time, its speed is $1.20 \times 10^{6} \mathrm{m} / \mathrm{s}$
(nonrelativistic, because $v$ is much less than the speed of light). (a) Find the acceleration of the proton. (b) How long does it take the proton to reach this speed? (c) How far has it moved in this time? (d) What is its kinetic energy at this time?

Key Concepts

Kinetic Energy

Kinetic energy is the energy a particle possesses due to its motion and is given by the formula KE = 1/2 mv^2. In problems involving acceleration by an electric field, the work done by the electric force is converted into kinetic energy, allowing one to determine the energy acquired by the particle as it accelerates from rest.

Kinematics Equations

Kinematics equations describe the motion of objects under constant acceleration. They are used to relate displacement, initial velocity, final velocity, time, and acceleration. In cases where a particle starts from rest and is subject to constant acceleration, these equations enable the calculation of the time taken to reach a given speed and the distance traveled.

Uniform Electric Field

A uniform electric field is one in which the electric field strength and direction are constant at every point in space. In such fields, a charged particle experiences a constant force, leading to uniform acceleration, which simplifies the analysis of particle motion and energy conversion.

Electric Force on a Charged Particle

The electric force acting on a charged particle in an electric field is given by the equation F = qE, where q is the charge of the particle and E is the magnitude of the electric field. This concept is crucial for understanding how particles like protons gain acceleration when subjected to external electric fields.

Newton's Second Law

Newton's Second Law of Motion, expressed as F = ma, connects the net force acting on an object with the acceleration produced. By combining this law with the electric force equation, one can determine the acceleration of a charged particle in an electric field, which is a fundamental step in analyzing its subsequent motion.

Transcript

00:01 In this problem we have four parts that we need to solve for the proton that is being accelerated by the given electric field so first we need to find its acceleration that is for the part of the problem we know that acceleration is force over a mass because from the second newton's law force equals mass times exploration in this case the force is due to the electric field and mass is corresponding to the proton and as a particle.

00:35 So from here we can say that the force, which is charge of the proton times electric field, is equal to acceleration times the mass of the proton.

00:53 And from here, we obtain that its duration of the proton is charge times electric field over the mass.

01:05 And we can substitute this value since they're known.

01:08 This is 1 .602 times 10 to the next negative 11 power times 640 over 1 .67 times 10 to the negative 27 power.

01:33 And this, when we put this one in the calculator, of course the dimensions are meters per second squared and when we put in place this one, it's in the calculator, we obtained that this equals to 6 .14 times 10 to the 10th power meters per second square and this is the answer for the part a of the problem.

01:58 Now for the part b of the problem, we know that since we need to find the time taken, the time that it takes for the protein to reach the given velocity, we can start from general kinematic equations which say that for a point mass under a constant acceleration because this is corresponding to the constant force of the electric field.

02:31 So the final velocity will be equal to the initial velocity plus acceleration times the time the past.

02:40 So from here you can say, when we can algebraically transform this expression to obtain the equation that we need, so t equals minus over a.

02:57 And we know this is in this acceleration from the part a of the problem so we can direct a substitute this is 1 .2 times 10 to the 6 power minus 0 since there is no initial velocity so we can just say without it and over 6 .14 times 10 to the 10th power and this is meters per second squared and this is meters per second so meters per second so centimeters cancel out, seconds and seconds squared...

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