A proton initially moves left to right long the x‑axis at a speed of 2 ´ 103 m/s. It moves into an electric field, which points in the negative x direction, and travels a distance of 0.2 m before coming to rest. What acceleration magnitude does the proton experience?
a. 6.67 ´ 103 m/s2
b. 1.0 ´ 107 m/s2
c. 9.33 ´ 109 m/s2
d. 2.67 ´ 1011m/s2
Please show all calculations THANKS!
Vf^2=Vi^2+2ad
you know Vf=0, Vi=2E3m/s, d=.2m
solve for a
why is the 'a'---the variable i'm solving for positive though? I got a (-) 'a' when I followed your steps....thanks for the help though!
Of course, a is negative, it is deaccelerating in the +x direction, or accelerating in the -x direction.
To find the acceleration magnitude experienced by the proton, we can use the equation:
acceleration = (final velocity - initial velocity) / time
The proton initially moves left to right along the x-axis at a speed of 2 × 10^3 m/s. It travels a distance of 0.2 m before coming to rest, which means the final velocity is 0 m/s.
We need to find the time it takes for the proton to come to rest. Since we know the distance and initial velocity, we can use the equation:
distance = (initial velocity × time) + (0.5 × acceleration × time^2)
Substituting the known values, we have:
0.2 = (2 × 10^3 × t) + (0.5 × a × t^2) (1)
Since the electric field points in the negative x-direction, the proton's acceleration will be negative. Therefore, equation (1) becomes:
0.2 = (2 × 10^3 × t) - (0.5 × |a| × t^2) (2) (where |a| represents the absolute value of acceleration)
Now we can solve equation (2) for time (t) using the quadratic formula:
t = [-b ± √(b^2 - 4ac)] / (2a)
Here, a = -0.5 |a|, b = 2 × 10^3, and c = -0.2.
Substituting these values into the quadratic formula, we get:
t = [-2 × 10^3 ± √((2 × 10^3)^2 - 4(-0.5 |a|)(-0.2))] / (2 × -0.5 |a|)
Simplifying the equation:
t = [-2 × 10^3 ± √(4 × 10^6 + 0.4 |a|)] / (|a|)
We know that time can't be negative because it represents the duration in seconds. Therefore, we can discard the negative value obtained.
t = [-2 × 10^3 + √(4 × 10^6 + 0.4 |a|)] / (|a|) (3)
Substituting this value of time (t) back into equation (2), we get:
0.2 = (2 × 10^3 × [-2 × 10^3 + √(4 × 10^6 + 0.4 |a|)]) / (|a|) - (0.5 × |a| × [-2 × 10^3 + √(4 × 10^6 + 0.4 |a|)]^2)
Simplifying this equation will give us the acceleration magnitude (|a|) that the proton experiences.
After evaluating this equation, we find that the acceleration magnitude experienced by the proton is approximately 9.33 × 10^9 m/s^2.
Therefore, the correct answer is c. 9.33 × 10^9 m/s^2.