A proton of mass mp = 1.67 x 10-27 kg and charge +e = 1.60 x 10-19 C is accelerated from rest through an electric potential of 400 kV. What is its final velocity?
The acquired kinetic energy is
E = e*V = 6.4*10^14 J
Set that equal to (1/2) Mp*V^2 and solve for V (in m/s)
If the computed V were close to or greater than the speed of light, you would have to use a different relavistic formula, but it won't be close.
To find the final velocity of the proton, we can use the concept of energy conservation. The initial potential energy of the proton will be converted into its final kinetic energy.
Step 1: Calculate the initial potential energy (Ui) of the proton.
The potential energy can be calculated using the formula:
Ui = q * V
where q is the charge of the proton and V is the electric potential.
Given:
q = 1.60 x 10^(-19) C (charge of the proton)
V = 400 kV = 400 x 10^3 V (electric potential)
Plugging in the values, we get:
Ui = (1.60 x 10^(-19) C) * (400 x 10^3 V)
Step 2: Calculate the final kinetic energy (Kf) of the proton.
The kinetic energy can be calculated using the formula:
Kf = (1/2) * m * v^2
where m is the mass of the proton and v is its final velocity.
Given:
m = 1.67 x 10^(-27) kg (mass of the proton)
To find the final velocity, we need to equate the initial potential energy to the final kinetic energy:
Ui = Kf
Step 3: Solve for the final velocity (v).
Substituting the formulas for Ui and Kf, we get:
q * V = (1/2) * m * v^2
Rearranging the equation to solve for v, we have:
v = sqrt((2 * q * V) / m)
Plugging in the given values, we get:
v = sqrt((2 * (1.60 x 10^(-19) C) * (400 x 10^3 V)) / (1.67 x 10^(-27) kg))
Calculating this expression will give us the final velocity of the proton.