An electron in the cathode ray tube of a tele-
vision set enters a region where it acceler-
ates uniformly from a speed of 53800 m/s to
a speed of 4.58 × 106 m/s in a distance of
2.82 cm.
What is its acceleration?
Answer in units of m/s2
Acceleration is the change in velocity over time:
Acceleration = (Final velocity - initial velocity)/time
Since there are no directions involved, speed is used instead of velocity.
So you know the initial speed:
Initial speed = 53800 m/s
And you know the final speed:
Final speed = 4.58 x 10^6 m/s
So, we can now do the speed part of the formula:
4.58 x 10^6 - 53800 = 4.53 x 10^6 m/s
This is the change that occurred in the distance 2.82 cm. So to find out how long that change took to occur we look at the following: (I will convert 2.82 cm to m = .0282 m)
0.0282 m * (1/4.53 x 10^6) s/m = 6.23 x 10^-9 s
So, now we have the initial speed, final speed, and the time so acceleration is:
Acceleration = (4.53 x 10^6 m/s)/(6.23 x 10^-9 s) = 7.27 x 10^14 (m/s)/s
To find the acceleration of the electron, we can use the formula for acceleration:
acceleration = (final velocity - initial velocity) / time
Given that the electron's initial velocity is 53800 m/s and its final velocity is 4.58 × 10^6 m/s, we need to determine the time it takes for this change in velocity to occur.
The information provided states that the electron accelerates uniformly over a distance of 2.82 cm. To convert this distance into meters, we divide by 100:
distance = 2.82 cm / 100 = 0.0282 m
Now, we need to find the time it takes for the electron to travel this distance. To do so, we can use the formula for uniformly accelerated motion:
distance = (initial velocity * time) + (0.5 * acceleration * time^2)
Since the electron starts from rest (initial velocity = 0 m/s), we can simplify the equation to:
distance = 0.5 * acceleration * time^2
Rearranging the equation to solve for time:
time^2 = (2 * distance) / acceleration
time = √((2 * distance) / acceleration)
Plugging in the values for distance (0.0282 m) and solving for time:
0.0282 = 0.5 * acceleration * time^2
time^2 = 0.0564 / acceleration
time = √(0.0564 / acceleration)
Now, we can substitute this value of time into the earlier formula for acceleration to solve for it:
acceleration = (final velocity - initial velocity) / time
acceleration = (4.58 × 10^6 m/s - 53800 m/s) / time
Substituting the value of time, we get:
acceleration = (4.58 × 10^6 m/s - 53800 m/s) / √(0.0564 / acceleration)
To solve this equation, we can simplify it by squaring both sides:
acceleration^2 = (4.58 × 10^6 - 53800)^2 / 0.0564
Taking the square root of both sides, we can find the value of acceleration:
acceleration = √((4.58 × 10^6 - 53800)^2 / 0.0564)
Calculating this expression, we find the value of acceleration.