⟨r²⟩ = 2 × (5 × 10⁻¹¹) × 2000 = 10⁻¹¹ × 2000 = 2 × 10⁻⁸ m².
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- A physics educator models the trajectory of a projectile launched at 30 m/s at 30° above horizontal. What is the maximum height reached? (Use g = 9.8 m/s², and ignore air resistance.)
- Vertical velocity = 30 × sin(30°) = 30 × 0.5 = 15 m/s. At maximum height, final vertical velocity = 0. Using v² = u² − 2gh → 0 = 15² − 2×9.8×h → 225 = 19.6h → h = 225 / 19.6 ≈ 11.48 m.
- A biologist collaborating with a physicist models diffusion of ions in a cell membrane. The mean squared displacement after t seconds is given by ⟨r²⟩ = 2Dt, where D = 5 × 10⁻¹¹ m²/s. What is ⟨r²⟩ after 2000 seconds?
- #### 2e-8
- An interactive simulation shows electrons moving through a magnetic field, deflecting by a centripetal force causing circular motion. If an electron moves at 2 × 10⁶ m/s perpendicular to a 0.5 Tesla field and has a charge of 1.6 × 10⁻¹⁹ C and mass 9.1 × 10⁻³¹ kg, what is the radius of its path?
- Centripetal force = qvB = mv²/r → r = mv / (qB). Plug in: r = (9.1 × 10⁻³¹ × 2 × 10⁶) / (1.6 × 10⁻¹⁹ × 0.5) = (1.82 × 10⁻²⁴) / (8 × 10⁻²⁰) = 2.275 × 10⁻⁵ m = 22.75 μm.
