First, we need to calculate the acceleration of the electron in the second stage of the particle accelerator:
We can use the equation v^2 = u^2 + 2as, where:
- v is the final velocity (3.25 x 10^6 m/s),
- u is the initial velocity (4.56 x 10^5 m/s),
- a is the acceleration,
- s is the distance (3.50 cm = 0.035 m).
Rearranging the equation to solve for acceleration:
a = (v^2 - u^2) / 2s
a = ((3.25 x 10^6)^2 - (4.56 x 10^5)^2) / (2 * 0.035)
a = (1.05625 x 10^13 - 2.0736 x 10^11) / 0.07
a ≈ 1.486 x 10^13 m/s^2
Next, we can calculate the time required for the electron to reach 99.9% of the speed of light:
Using the equation v = u + at, where:
- v is the final velocity (2.997 x 10^8 m/s),
- u is the initial velocity (3.25 x 10^6 m/s),
- a is the acceleration (1.486 x 10^13 m/s^2),
- t is the time.
Rearranging the equation to solve for time:
t = (v - u) / a
t = (2.997 x 10^8 - 3.25 x 10^6) / 1.486 x 10^13
t ≈ 19.94 × 10^-5 seconds
Therefore, the magnitude of acceleration required for the electron to reach 99.9% of the speed of light in the second stage is approximately 1.486 x 10^13 m/s^2, and the time required is approximately 19.94 × 10^-5 seconds.
An electron with an initial speed of 4.56 x 10^5m/s enters the second section of a particle accelerator that is 3.50 cm long. In this section, the electron is accelerated to a speed of 3.25 x 10^6 m/s.
Particle accelerators are known to increase the velocity of a particle to 99.9% of the speed of light (2.997 x10^8m/s). What magnitude of acceleration and time is required for the electron to reach this velocity in the second stage?
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