A ball weighing 10 g is moving with a velocity of 90 ms⁻¹. If the uncertainty in its velocity is 5%, then the uncertainty in its position is ___________ × 10⁻³³ m. (Rounded off to the nearest integer)
[Given : h = 6.63 × 10⁻³⁴ Js]

Solution

Chemistry ma’am · 3 steps

Find Δ v from the 5% first, then put Δ p = mΔ v into Heisenberg's relation.

1. Uncertainty in velocity. 5% of 90 m s⁻¹ is Δ v = 0.05 × 90 = 4.5 m s⁻¹. The mass in SI units is m = 10 g = 0.01 kg.

   Δ x = h/(4π m Δ v)

2. Put in numbers. mΔ v = 0.01 × 4.5 = 0.045 kg m s⁻¹, so 4π m Δ v = 4 × 3.1416 × 0.045 = 0.5655. Then Δ x = (6.63 × 10⁻³⁴)/0.5655 = 1.17 × 10⁻³³ m.
3. So. Δ x ≈ 1.17 × 10⁻³³ m, which rounds to 1.

The common slip: 15. Leaving out the 4π and using Δ x = h/(mΔ v) gives 1.47 × 10⁻³² m, which reads as 15, not 1.

The idea to keep. For Heisenberg questions, turn the percentage into an actual Δ v, keep mass in kg, and use Δ x · mΔ v = h/4π. Do not forget the 4π.

Answer from the NTA final answer key. Solution by PrepWiser.