A proton has an initial speed of 3.9×105 m/s.
What potential difference is required to bring the proton to rest?
What potential difference is required to reduce the initial speed of the proton by a factor of 2?
What potential difference is required to reduce the initial kinetic energy of the proton by a factor of 2?
To find the potential difference required to bring the proton to rest, we can use the concept of conservation of energy. The initial kinetic energy of the proton can be equated to the change in potential energy when the proton is brought to rest. The equation is:
Initial kinetic energy = final potential energy
The initial kinetic energy of the proton is given by:
KE_initial = (1/2)mv^2
where m is the mass of the proton and v is the initial speed.
To bring the proton to rest, its final kinetic energy will be zero, so the final potential energy is equal to the initial kinetic energy. Therefore:
Final potential energy = KE_initial = (1/2)mv^2
Now, we need to calculate the potential difference associated with this change in potential energy. The potential difference (V) is defined as the change in potential energy (ΔU) per unit charge (q): V = ΔU/q.
For a proton, the charge (q) is the elementary charge (e) or 1.6 × 10^-19 C. Therefore, the potential difference required to bring the proton to rest is:
V = ΔU/q = (KE_initial)/q = (1/2)mv^2/q
To find the numerical value, you need to know the mass of a proton. The mass of a proton is approximately 1.67 × 10^-27 kg. Plugging in the values:
V = (1/2)(1.67 × 10^-27 kg)(3.9 × 10^5 m/s)^2/(1.6 × 10^-19 C)
Simplifying the equation gives the potential difference required to bring the proton to rest.
To find the potential difference required to reduce the initial speed of the proton by a factor of 2, we can use the principle of conservation of energy again. This time, the final kinetic energy is (1/2)mv_final^2, where v_final is the final velocity (half of the initial velocity).
Using the same equation:
Final potential energy = KE_initial = (1/2)mv^2
The potential difference (V) needed is:
V = ΔU/q = (KE_initial - KE_final)/q = [(1/2)mv^2 - (1/2)mv_final^2]/q
Substituting the values:
V = [(1/2)(1.67 × 10^-27 kg)(3.9 × 10^5 m/s)^2 - (1/2)(1.67 × 10^-27 kg)((3.9 × 10^5 m/s)/2)^2]/(1.6 × 10^-19 C)
Similarly, to find the potential difference required to reduce the initial kinetic energy of the proton by a factor of 2, simply substitute the new kinetic energy (KE_final) into the formula for potential difference:
V = [(1/2)mv^2 - (1/2)(1/2)mv^2]/q
where the initial kinetic energy is (1/2)mv^2 and the final kinetic energy is (1/2)(1/2)mv^2.