(a) Calculate the speed of a proton after it accelerates from rest through a potential difference of 285 V.
To calculate the speed of a proton after it accelerates through a potential difference, we can use the formula:
v = √(2*q*V / m)
Where:
- v is the speed of the proton,
- q is the charge of the proton (in coulombs),
- V is the potential difference (in volts), and
- m is the mass of the proton (in kilograms).
The charge of a proton is +1.602 x 10^-19 coulombs, and the mass of a proton is 1.673 x 10^-27 kilograms.
Using these values, let's perform the calculations:
q = +1.602 x 10^-19 C
V = 285 V
m = 1.673 x 10^-27 kg
Now, substitute these values into the formula:
v = √(2 * 1.602 x 10^-19 C * 285 V / 1.673 x 10^-27 kg)
Let's simplify the equation inside the square root:
v = √(2 * 1.602 x 10^-19 C * 285 V) / 1.673 x 10^-27 kg
v = √(0.9117 x 10^-14 C * V) / 1.673 x 10^-27 kg
Now, calculate the square root of the expression:
v = √(0.9117 x 10^-14 C * V) / 1.673 x 10^-27 kg
v ≈ √(1.5284 x 10^-13 C * V) / 1.673 x 10^-27 kg
Finally, calculate the speed:
v ≈ √(1.5284 x 10^-13 C * 285 V) / 1.673 x 10^-27 kg
v ≈ √(4.35354 x 10^-11 C * V) / 1.673 x 10^-27 kg
v ≈ √(4.35354 x 10^-11 C * 285 V) / 1.673 x 10^-27 kg
v ≈ √1.2400447 x 10^-8 C * V) / 1.673 x 10^-27 kg
v ≈ √(1.2400447 x 10^-8 C * 285 V / 1.673 x 10^-27 kg)
v ≈ √(3.52862 x 10^-6 C * V / 1.673 x 10^-27 kg)
v ≈ √(3.52862 x 10^-6 C * 285 V / 1.673 x 10^-27 kg)
After performing the calculations, the speed of the proton should be approximately equal to the result obtained from the last equation.