An electron starting from rest acquires 4.20 keV of kinetic energy in moving from point A to point B. How much kinetic energy would a proton acquire, starting from rest at B and moving to point A?
Determine the ratio of their speeds at the end of their respective trajectories.
To find the kinetic energy acquired by a proton, we can use the fact that the kinetic energy is directly proportional to the mass and the square of the velocity. The mass of a proton is approximately 1836 times the mass of an electron.
We are given that the electron acquires 4.20 keV of kinetic energy in moving from point A to point B. To find the kinetic energy acquired by the proton, we can use the ratio of their masses.
Let's calculate the kinetic energy acquired by the electron and then use the mass ratio to find the kinetic energy acquired by the proton.
1. Calculate the kinetic energy acquired by the electron:
Given: Kinetic energy of the electron = 4.20 keV
We know that the kinetic energy (KE) of an object is given by the formula:
KE = (1/2) * m * v^2
where m is the mass and v is the velocity.
Since the electron starts from rest, its initial velocity (v1) is 0. So the kinetic energy (KE1) acquired by the electron is:
KE1 = (1/2) * m(electron) * v1^2
= (1/2) * m(electron) * 0^2
= 0
The final kinetic energy (KE2) acquired by the electron is then:
KE2 = 4.20 keV
2. Calculate the kinetic energy acquired by the proton:
Since the proton starts from rest at point B, its final kinetic energy (KE(proton)) is equal to the kinetic energy acquired by the electron (KE2), multiplied by the mass ratio of the proton to the electron.
The mass ratio (m(proton) / m(electron)) is approximately 1836.
KE(proton) = KE2 * (m(proton) / m(electron))
3. Calculate the ratio of their speeds at the end of their respective trajectories:
Let's assume the final velocities of the electron and the proton are v2(electron) and v2(proton) respectively.
To find the ratio of their speeds, we can use the fact that kinetic energy is directly proportional to the square of the velocity. So we have:
KE2(electron) / KE2(proton) = (v2(electron))^2 / (v2(proton))^2
Since we know the values of KE2(electron) and KE2(proton), we can solve for the ratio of their speeds.
Note: To make the calculation easier, we can cancel out keV (kilo-electron volts) by converting it to joules (J) using the conversion factor: 1 keV = 1.602 x 10^-16 J.
To find the kinetic energy gained by a proton, starting from rest at point B and moving to point A, we can use the principle of conservation of energy.
The electron gains 4.20 keV of kinetic energy, so we know that the initial potential energy at point A must be equal to the final kinetic energy at point B for the electron.
Using the formula for kinetic energy:
K.E. = (1/2)mv²,
where K.E. is the kinetic energy acquired, m is the mass of the particle, and v is the velocity of the particle.
Since the electron already acquired 4.20 keV of kinetic energy, we have:
(1/2)m(electron)v(electron)² = 4.20 keV
To find the kinetic energy gained by the proton, we can use the same formula:
(1/2)m(proton)v(proton)² = K.E. (proton)
Now, let's find the ratio of their speeds at the end of their respective trajectories.
The kinetic energy gained by the proton is equal to the initial potential energy at point A for the electron, so we can equate the two equations:
(1/2)m(electron)v(electron)² = (1/2)m(proton)v(proton)²
Dividing both sides of the equation by (1/2)m(proton), we get:
v(electron)² = v(proton)²
Taking the square root of both sides, we have:
v(electron) = v(proton)
Therefore, the ratio of their speeds at the end of their respective trajectories is 1:1.
I assume that the energy comes from asn E field between A and B.
The force is proportional to charge and E so the force magnitude is equal (opposite sign though )
The distance is the same so the work done is the same.
So the kinetic energy is the same (1/2) m v^2
so
(1/2) Mproton Vproton^2 = (1/2) Melectron Velectron^2
so
Vproton^2/Velectron^2 = Melectron/Mproton