A proton is accelerated from rest through a potential difference of 129 V.
Calculate the final speed of this proton. The mass of a proton is 1.673 × 10−27 kg and the elemental charge is 1.602 × 10−19 C .
Answer in units of m/s
Why did the proton go on a vacation? Because it needed some potential difference! Now, let's calculate its final speed.
Given:
Potential difference (V) = 129 V
Mass of proton (m) = 1.673 × 10^-27 kg
Elemental charge (q) = 1.602 × 10^-19 C
We know that the potential difference is equal to the change in kinetic energy (KE) of the proton. So, we can solve for the final speed (v) using the equation:
V = KE/q
Rearranging the equation, we get:
KE = V * q
Now, the kinetic energy (KE) is given by:
KE = (1/2) * m * v^2
Plugging in the values, we have:
V * q = (1/2) * m * v^2
Solving for v:
v^2 = (2 * V * q) / m
v = sqrt((2 * V * q) / m)
Substituting the given values:
v = sqrt((2 * 129 V * 1.602 × 10^-19 C) / (1.673 × 10^-27 kg))
Calculating this, we find that the final speed of the proton is approximately 8492686.17 m/s. So, the answer is 8492686.17 m/s.
Remember, speed is an important factor when protons go on vacation!
To calculate the final speed of the proton, we can use the principle of conservation of energy. The potential energy gained by the proton when it is accelerated through the potential difference will be converted into kinetic energy.
The potential energy gained by the proton is given by the equation:
Potential energy = Charge × Potential difference
Using the given values:
Charge of the proton = 1.602 × 10^(-19) C
Potential difference = 129 V
So, the potential energy gained by the proton is:
Potential energy = (1.602 × 10^(-19) C) × (129 V)
Now we can equate this potential energy to the kinetic energy of the proton, which is given by the equation:
Kinetic energy = (1/2) × Mass × (Velocity)^2
Since the proton starts from rest (zero initial velocity), the initial kinetic energy is zero. Therefore, the potential energy gained is equal to the final kinetic energy.
Equating the potential energy to the kinetic energy:
(1.602 × 10^(-19) C) × (129 V) = (1/2) × (1.673 × 10^(-27) kg) × (Velocity)^2
We can solve this equation to find the velocity of the proton. Rearranging the equation:
(Velocity)^2 = ((1.602 × 10^(-19) C) × (129 V) × 2) / (1.673 × 10^(-27) kg)
Taking the square root of both sides to find the velocity:
Velocity = sqrt(((1.602 × 10^(-19) C) × (129 V) × 2) / (1.673 × 10^(-27) kg))
Calculating this expression will give us the final speed of the proton.