In a 100 m linear accelerator, an electron is accelerated to 1.40% of the speed of light in 46.0 m before it coasts for 54.0 m to a target. What's the electron's acceleration during the first 46.0 m?
b) How long does the total flight take?
To find the electron's acceleration during the first 46.0 m, we can use the equation for acceleration:
acceleration = change in velocity / change in time
Given that the electron is accelerated to 1.40% of the speed of light, we can calculate the change in velocity as follows:
change in velocity = 1.40% of the speed of light
Now, we need to determine the time it takes for the electron to travel the first 46.0 m. Since we don't have this information directly, we can use the following equation to find the time:
distance = (initial velocity * time) + (1/2 * acceleration * time^2)
In this case, since the initial velocity is 0 (electron starts at rest), the equation simplifies as follows:
distance = (1/2 * acceleration * time^2)
Now, we can use this equation to find the time it takes to travel the first 46.0 m.
Let's go step by step.
Step 1: Calculate the change in velocity.
change in velocity = 1.40% * speed of light
Step 2: Calculate the time it takes to travel the first 46.0 m.
46.0 = (1/2 * acceleration * time^2)
Now we have an equation with two unknowns, acceleration and time. To solve it, we need a second equation. We know that the electron reaches 1.40% of the speed of light in a distance of 46.0 m, so we can use the equation for velocity:
v = u + at
Where:
v = final velocity (1.40% of the speed of light)
u = initial velocity (0)
a = acceleration (unknown)
t = time (unknown, but this is what we want to find)
Step 3: Calculate the time it takes to accelerate to 1.40% of the speed of light.
1.40% of the speed of light = 0 + acceleration * time
Now we have two equations to solve for acceleration and time. Let's proceed to solve them simultaneously.
Step 4: Solve the equations simultaneously to find acceleration and time.
Using the two equations we derived in Step 2 and Step 3, we can solve them simultaneously to find the values of acceleration and time.
acceleration * time = 1.40% of the speed of light -- (Equation 1)
(1/2 * acceleration * time^2) = 46.0 -- (Equation 2)
Simplifying Equation 1, we get:
acceleration = (1.40% of the speed of light) / time
Substituting this value into Equation 2:
(1/2 * (1.40% of the speed of light) * time^2) = 46.0
Simplifying further:
(0.7% of the speed of light * time^2) = 46.0
Now, we can solve this equation to find the value of time.
Step 5: Solve the equation to find the value of time.
0.7% of the speed of light * time^2 = 46.0
Dividing both sides by 0.7% of the speed of light:
time^2 = 46.0 / (0.7% of the speed of light)
Now, let's plug in the values and calculate the time.
Step 6: Calculate the value of time.
time^2 = 46.0 / (0.007 * speed of light)
Now, we can take the square root of both sides to find the value of time.
Step 7: Calculate the square root of the right-hand side to find the value of time.
time = √(46.0 / (0.007 * speed of light))
After finding the value of time, you can evaluate it to get the acceleration during the first 46.0 m.
To find the total flight time, we can use the equation:
total time = time to accelerate + time to coast
Given that the electron coasts for 54.0 m, we know that the time to coast is equal to:
time to coast = distance to coast / final velocity
Now, we can find the total flight time by adding the time to accelerate and the time to coast.
I hope this step-by-step explanation helps you to understand the process. Let me know if you have any further questions!
To find the electron's acceleration during the first 46.0 m of the linear accelerator, we can use the equation of motion, which relates the final velocity (v), initial velocity (u), acceleration (a), and distance (s). The equation can be written as:
v^2 = u^2 + 2as
In this case, we know the final velocity (v) is 1.40% of the speed of light, so we can convert it to meters per second. The speed of light is 3 x 10^8 m/s, so 1.40% of that is:
v = (1.40/100) * (3 x 10^8) m/s = 4.2 x 10^6 m/s
The initial velocity (u) is 0 because the electron starts from rest. The distance (s) is given as 46.0 m. Now we can rearrange the equation to solve for acceleration (a):
a = (v^2 - u^2) / (2s)
Substituting the known values:
a = (4.2 x 10^6)^2 / (2 * 46.0)
Now, let's calculate this:
a = 1.79 x 10^11 m/s^2
Therefore, the electron's acceleration during the first 46.0 m of the linear accelerator is 1.79 x 10^11 m/s^2.
Moving on to the second part of the question, to find the total flight time, we need to consider two parts: the time it takes for the electron to accelerate (46.0 m) and the time it takes for it to coast (54.0 m). Since we know the distance and the acceleration, we can use the following equation:
s = ut + (1/2)at^2
For the acceleration phase, the initial velocity (u) is 0, and the distance (s) is 46.0 m. Rearranging the equation to solve for time (t):
t = √(2s / a)
Plugging in the known values:
t = √(2 * 46.0 / 1.79 x 10^11)
Calculating this:
t ≈ 3.005 x 10^-7 s
So, the time it takes for the electron to accelerate is approximately 3.005 x 10^-7 seconds.
Next, let's calculate the time it takes for the electron to coast for 54.0 m. During this phase, there is no acceleration, so the equation becomes:
s = ut
The initial velocity (u) is the final velocity attained during the acceleration phase, which is 4.2 x 10^6 m/s. The distance (s) is given as 54.0 m. Rearranging the equation:
t = s / u
Plugging in the known values:
t = 54.0 / (4.2 x 10^6)
Calculating this:
t ≈ 1.286 x 10^-5 s
Therefore, the time it takes for the electron to coast for 54.0 m is approximately 1.286 x 10^-5 seconds.
To find the total flight time, we need to add the time for acceleration and the time for coasting:
Total flight time ≈ Acceleration time + Coast time
Total flight time ≈ 3.005 x 10^-7 + 1.286 x 10^-5 s
Calculating this:
Total flight time ≈ 1.317 x 10^-5 s
Therefore, the total flight time is approximately 1.317 x 10^-5 seconds.