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 Its initial speed.
in the simplest steps
To determine the initial speed of the proton, we can use the equation of motion for an object in a uniform acceleration:
vf^2 = vi^2 + 2ax
Where vf is the final velocity (0 m/s since the proton comes to rest), vi is the initial velocity (unknown), a is the acceleration (caused by the electric field), and x is the displacement (7.00 cm or 0.07 m).
Given that the electric field E = -6x10^51 N/C, we can calculate the acceleration of the proton using Newton's second law:
F = ma
Where F is the force acting on the proton and m is the mass of the proton. The force acting on the proton is given by:
F = qE
Where q is the charge of the proton. The charge of the proton is the elementary charge e:
q = e = 1.6x10^-19 C
Substituting this into the equation:
Fe = ma
qE = ma
Solving for the acceleration:
a = qE/m = (1.6x10^-19 C)(-6x10^51 N/C)/(1.67x10^-27 kg) = -9.6x10^-7 m/s^2
Substituting the known values into the equation of motion:
(0 m/s)^2 = vi^2 + 2(-9.6x10^-7 m/s^2)(0.07 m)
0 = vi^2 - 1.344x10^-8 m^2/s^2
Solving for vi:
vi^2 = 1.344x10^-8 m^2/s^2
vi = sqrt(1.344x10^-8 m^2/s^2) ≈ 3.6667x10^-4 m/s
Therefore, the initial speed of the proton is approximately 3.67x10^-4 m/s.
To determine the initial speed of the proton, you can use the kinematic equation for acceleration in one dimension:
v^2 = u^2 + 2ad
Where:
v = final velocity (which is 0 m/s, as the proton comes to rest)
u = initial velocity (which we need to find)
a = acceleration (which is the electric field in this case)
d = distance traveled (which is given as 7.00 cm)
Let's plug in the given values and solve for u:
0 = u^2 + 2(-6x10^5)(7.00x10^-2)
Multiply the acceleration by the distance:
0 = u^2 - 8.4x10^4
Rearrange the equation to solve for u^2:
u^2 = 8.4x10^4
Take the square root of both sides to find u:
u = √(8.4x10^4)
Now, simply calculate the square root to find the solution:
u ≈ 290.47 m/s
So, the initial speed of the proton is approximately 290.47 m/s.
To determine the initial speed of the proton, we can use the principle of conservation of energy. The work done by the electric field can be equated to the change in kinetic energy of the proton as it comes to rest.
1. Identify the given values:
- Initial position (At1 = 0)
- Electric field (E = -6x10^5 N/C)
- Final position (Af2 = 7.00 cm = 0.07 m)
2. Calculate the work done by the electric field:
Work (W) = force (F) * displacement (d)
Since the electric field is uniform, the force experienced by the proton (F) is equal to the charge of the proton (q) multiplied by the electric field strength (E).
W = F * d = q * E * d
Since the charge of a proton is q = +1.6x10^-19 C, we can substitute the values and calculate the work done.
3. Calculate the change in kinetic energy:
The work done by the electric field (W) is equal to the change in kinetic energy (∆KE) of the proton.
∆KE = W
Rearrange the equation to solve for the initial kinetic energy (KE1):
KE1 = ∆KE + KE2
Since the final kinetic energy (KE2) is zero when the proton comes to rest, we have:
KE1 = W + 0
Therefore, the initial kinetic energy is equal to the work done by the electric field.
4. Calculate the initial speed:
The initial kinetic energy can be related to the initial speed (v1) using the equation:
KE1 = (1/2) * m * v1^2
Rearrange the equation to solve for the initial speed:
v1 = √(2 * KE1 / m)
The mass of a proton is m = 1.67x10^-27 kg. By substituting the values, we can calculate the initial speed.
By following these steps, you can determine the initial speed of the proton using the given information and the principles of conservation of energy.