At1=0, a proton is projected in the positive x-direction into a region of a uniform electric field of E=-6x1051. The proton travels 7.00 cm as it comes to rest. Determine
a) The acceleration of the proton.
b) Its initial speed,
c) The time interval over which the proton comes to rest
in simple steps
To find the answers to the questions, we can use the kinematic equations of motion.
a) The acceleration of the proton can be found using the equation:
vf^2 = vi^2 + 2aΔx
Since the proton comes to rest, vf = 0. Plugging in the values, we get:
0 = vi^2 + 2a(0.07)
Simplifying the equation, we get:
0 = vi^2 + 0.14a
Since vi = 0 (the proton starts from rest), the equation becomes:
0 = 0 + 0.14a
Simplifying further, we find:
0 = 0.14a
Therefore, the acceleration of the proton is zero.
b) The initial speed of the proton can be found using the equation:
vf = vi + at
Since the acceleration is zero, the equation becomes:
vf = vi
Therefore, the initial speed of the proton is equal to the final speed, which is zero.
c) The time interval over which the proton comes to rest can be found using the equation:
vf = vi + at
Since the acceleration is zero, the equation becomes:
0 = vi + 0t
Simplifying, we find:
0 = vi
Therefore, the time interval over which the proton comes to rest is also zero.
In summary:
a) The acceleration of the proton is zero.
b) The initial speed of the proton is zero.
c) The time interval over which the proton comes to rest is zero.
To solve this problem, we can use the equations of motion for uniformly accelerated motion.
a) The acceleration of the proton can be determined using the equation:
v^2 = u^2 + 2as
where,
v = final velocity (0 m/s, as it comes to rest)
u = initial velocity (to be determined)
a = acceleration (to be determined)
s = distance traveled (7.00 cm = 0.07 m)
Rearranging the equation and substituting the given values, we have:
0^2 = u^2 + 2a(0.07)
Simplifying the equation, we have:
0 = u^2 + 0.14a (equation 1)
b) The initial speed of the proton can be determined using the equation:
v = u + at
Substituting the given values, we have:
0 = u + a(t) (equation 2)
c) The time interval over which the proton comes to rest can be determined by substituting the given values into equation 2 and solving for t.
Let's solve each part step by step:
Step 1: Solve equation 1 for a
From equation 1, we have:
0 = u^2 + 0.14a
Rearranging the equation, we get:
a = -u^2 / 0.14 (equation 3)
Step 2: Substitute equation 3 into equation 2
From equation 2, we have:
0 = u + a(t)
Substituting the value of a from equation 3, we get:
0 = u - u^2 / 0.14 * t
Simplifying the equation, we have:
t = u / (u / 0.14)
Step 3: Determine the unknowns
a) The acceleration of the proton (a) can be determined using equation 3:
a = -u^2 / 0.14
b) The initial speed of the proton (u) can be determined from equation 1:
0 = u^2 + 0.14a
Solving this equation for u, we get two possible values. However, since we are dealing with a proton, the initial speed (u) cannot be negative. Therefore, the initial speed (u) is:
u = √( -0.14a)
c) The time interval over which the proton comes to rest (t) can be determined using step 2:
t = u / (u / 0.14)
To solve this problem, we can use the equations of motion to determine the acceleration, initial speed, and time interval. Here are the steps to calculate each value:
a) Acceleration (a):
Since the proton comes to rest, its final velocity (v_f) is 0. We can use the equation:
v_f^2 = v_i^2 + 2aΔx
Where:
- v_f is the final velocity (0 in this case)
- v_i is the initial velocity (unknown)
- a is the acceleration (to be determined)
- Δx is the displacement (7.00 cm or 0.07 m)
Rearranging the equation, we get:
a = - v_i^2 / 2Δx
Plugging in the given values, we have:
a = - v_i^2 / 2(0.07)
b) Initial speed (v_i):
To find the initial speed, we can use the equation of motion:
v_f = v_i + at
Since the final velocity (v_f) is 0, the equation becomes:
0 = v_i + at
We can isolate v_i:
v_i = -at
Plugging in the known values, we get:
v_i = -a(t)
c) Time interval (t):
To find the time interval, we can rearrange the equation from step b:
t = -v_i / a
Once we have determined the values of a and v_i using the steps above, we can calculate the time interval.
Now, let's solve each step one by one:
a) Acceleration (a):
Using the equation a = - v_i^2 / 2Δx, we substitute the values:
a = -(v_i^2) / (2Δx)
a = -(v_i^2) / (2 * 0.07)
b) Initial speed (v_i):
Using the equation v_i = -at, we substitute the value of a from step a:
v_i = -a(t)
c) Time interval (t):
Using the equation t = -v_i / a, we substitute the values of v_i and a obtained from steps a and b.
After calculating the values of a, v_i, and t using the above steps, we will have the answers to each part of the question.