Question

The velocity of a proton in an accelerator is known to an accuracy of 0.250% of the speed of light. (This could be small compared with its velocity.) What is the smallest possible uncertainty in its position?

a) 2.58 x 10^(-16) m
b) 3.92 x 10^(-16) m
c) 5.17 x 10^(-16) m
d) 6.81 x 10^(-16) m

Answer 1

The smallest possible uncertainty in the proton's position is 7.50 × 10^16 m.

Step-by-step explanation:

The uncertainty principle in quantum mechanics states that there is a limit to how accurately the position and velocity of a particle can be simultaneously measured. The uncertainty in position (Δx) is related to the uncertainty in velocity (Δv) by the equation ΔxΔv ≥ h/(4π), where h is the Planck constant. In this case, the uncertainty in velocity is given as 0.250% of the speed of light. To find the smallest possible uncertainty in position, we can use the formula Δx = (Δv)(c), where c is the speed of light.

Plugging in the values, Δx = (0.00250)(3.00 × 10^8 m/s) = 7.50 × 10^5 m, which is equivalent to 7.50 × 10^5 × 10^9 pm = 7.50 × 10^14 pm. Therefore, the smallest possible uncertainty in its position is 7.50 × 10^14 pm, which is approximately 7.50 × 10^16 m.