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 simple steps
To determine the initial speed of the proton, we can use the equation:
vf^2 = vi^2 + 2ad,
where:
- vf represents the final velocity, which is zero since the proton comes to rest,
- vi represents the initial velocity (what we want to find),
- a represents the acceleration of the proton,
- d represents the distance traveled by the proton.
First, let's find the acceleration of the proton using the electric field.
The force experienced by the proton due to the electric field is given by:
F = qE,
where:
- F represents the force experienced by the proton,
- q represents the charge of the proton,
- E represents the electric field strength.
The motion of the proton can be considered 1D, so the force can be related to the acceleration using Newton's second law:
F = ma,
where:
- m represents the mass of the proton,
- a represents the acceleration.
Since the problem doesn't provide the mass of the proton, we'll use the known value: m = 1.67 × 10^(-27) kg
Combining the equations for force, acceleration, and electric field:
qE = ma.
Solving for 'a':
a = qE / m.
The charge of a proton is given by q = 1.6 × 10^(-19) C.
Substituting the values:
a = (1.6 × 10^(-19) C)(-6 × 10^5 N/C) / (1.67 × 10^(-27) kg).
Now, let's calculate 'a':
a ≈ -5.76 × 10^13 m/s².
Now that we have the acceleration, we can use the motion equation to find the initial velocity.
vf^2 = vi^2 + 2ad.
Since vf is zero:
0 = vi^2 + 2ad.
Rearranging the equation to solve for 'vi':
vi^2 = -2ad.
Taking the square root of both sides:
vi = √(-2ad).
Substituting the values:
vi = √(-2)(-5.76 × 10^13 m/s²)(0.07 m).
Solving for 'vi':
vi ≈ √(8.064 × 10^12 m²/s²).
Simplifying:
vi ≈ 2.84 × 10^6 m/s.
Therefore, the initial speed of the proton is approximately 2.84 × 10^6 m/s.
To determine the initial speed of the proton, we can use the following steps:
Step 1: Determine the work done on the proton by the electric field.
The work done (W) by the electric field on the proton is given by the formula:
W = q * ΔV
where q is the charge of the proton and ΔV is the change in electric potential.
Step 2: Calculate the change in electric potential (ΔV).
Since the electric field is uniform in the x-direction, the change in electric potential (ΔV) can be calculated using the following formula:
ΔV = -E * d
where E is the electric field and d is the distance or displacement in the x-direction.
Step 3: Calculate the work done (W).
Substituting the values for q, ΔV, and E obtained from the problem into the equation W = q * ΔV, we can calculate the work done on the proton.
Step 4: Relate work done to the change in kinetic energy.
The work done on the proton is equal to the change in its kinetic energy (ΔKE), given by the formula:
ΔKE = W
Step 5: Calculate the initial kinetic energy (KE_initial).
The initial kinetic energy (KE_initial) of the proton can be calculated by subtracting the change in kinetic energy (ΔKE) from the final kinetic energy (KE_final), where the final kinetic energy is zero since the proton comes to rest.
KE_final = 0
Step 6: Convert the initial kinetic energy (KE_initial) to the initial speed (v_initial).
The initial kinetic energy (KE_initial) can be related to the initial speed (v_initial) by the formula:
KE_initial = (1/2) * m * v_initial^2
where m is the mass of the proton.
Step 7: Rearrange the equation to solve for v_initial.
Rearranging the equation KE_initial = (1/2) * m * v_initial^2 allows us to solve for the initial speed (v_initial).
Now, let's calculate each step in detail.
Step 1: Determine the work done on the proton by the electric field.
The charge of a proton is given by q = e, where e is the elementary charge: e = 1.602 × 10^(-19) C. However, in this calculation, the charge q is positive, so we can use q = +1.602 × 10^(-19) C.
Step 2: Calculate the change in electric potential (ΔV).
The electric field is given by the equation E = -6 × 10^5 N/C (negative sign indicates the direction opposite to the positive x-direction), and the distance traveled by the proton is given by d = 7.00 cm = 0.07 m.
ΔV = -E * d
ΔV = (-6 × 10^5 N/C) * (0.07 m)
Step 3: Calculate the work done (W).
Using the formula W = q * ΔV, where q = 1.602 × 10^(-19) C, we can calculate the work done on the proton.
W = (1.602 × 10^(-19) C) * ΔV
Step 4: Relate work done to the change in kinetic energy.
The work done (W) is equal to the change in kinetic energy (ΔKE).
ΔKE = W
Step 5: Calculate the initial kinetic energy (KE_initial).
Since the final kinetic energy (KE_final) is zero, the initial kinetic energy (KE_initial) is equal to the negative change in kinetic energy (ΔKE).
KE_initial = -ΔKE
Step 6: Convert the initial kinetic energy (KE_initial) to the initial speed (v_initial).
Using the equation KE_initial = (1/2) * m * v_initial^2, where m is the mass of the proton (m = 1.67 × 10^(-27) kg), we can calculate the initial speed (v_initial).
KE_initial = (1/2) * m * v_initial^2
Step 7: Rearrange the equation to solve for v_initial.
To solve for the initial speed (v_initial), we rearrange the equation to isolate v_initial:
v_initial^2 = (2 * KE_initial) / m
v_initial = sqrt((2 * KE_initial) / m)
Now, we can substitute the values we previously obtained into the equation, v_initial = sqrt((2 * KE_initial) / m), to calculate the initial speed of the proton.
To determine the initial speed of the proton, we can use the equation of motion for an object in an electric field:
v^2 = u^2 + 2aΔx
where:
v = final velocity (which is 0 m/s since the proton comes to rest)
u = initial velocity (what we need to find)
a = acceleration due to the electric field (qE/m, where q is the charge and m is the mass)
Δx = distance traveled (given as 7.00 cm = 0.07 m)
Step 1: Find the acceleration of the proton
We know the charge of a proton (q) is +1.6 x 10^-19 C and its mass (m) is 1.67 x 10^-27 kg. Substitute these values into the equation:
a = (qE) / m
a = (1.6 x 10^-19 C) * (-6 x 10^51 N/C) / (1.67 x 10^-27 kg)
a ≈ -5.76 x 10^26 m/s^2
Step 2: Use the equation of motion to find the initial velocity
Now we have all the values we need to solve for the initial velocity:
0^2 = u^2 + 2(-5.76 x 10^26 m/s^2)(0.07 m)
Simplifying the equation:
0 = u^2 + (-2.03 x 10^25 m^2/s^2)
Since the final velocity is zero, the equation becomes:
u^2 = 2.03 x 10^25 m^2/s^2
Taking the square root of both sides:
u ≈ ±4.51 x 10^12 m/s
Since velocity is a vector quantity, we need to consider only the positive value:
u ≈ 4.51 x 10^12 m/s
Therefore, the initial speed of the proton is approximately 4.51 x 10^12 m/s.