Question

Two balls each of 5 cm in diameter are placed 200 m apart on a horizontal frictionless plane at 45° N. The balls are impulsively propelled directly at each other with equal speeds. What must be the speed in m/s so that the two balls just miss each other? (Take $\Omega = 7.29 \times 10^{-5} \, \text{rad/s}$, rounded off to two decimal places).

Hint:

Coriolis deflection depends on travel time. Slower speed → greater deflection, faster speed → less deflection.

Answer 1

Correct Answer: 20.6

Step 1: Deflection due to Coriolis force.
The Coriolis acceleration is: \[ a_c = 2 \Omega v \sin\phi \] At latitude $\phi = 45^\circ$: \[ a_c = 2 \times 7.29 \times 10^{-5} \times v \times \frac{\sqrt{2}}{2} = 1.03 \times 10^{-4} v \]
Step 2: Lateral deflection after traveling half the distance.
Time to travel half distance (100 m each): \[ t = \frac{100}{v} \] Lateral deflection (Coriolis): \[ d = \frac{1}{2} a_c t^2 = \frac{1}{2} (1.03 \times 10^{-4} v) \left(\frac{100}{v}\right)^2 \] \[ d = \frac{0.5 \times 1.03 \times 10^{-4} \times 10000}{v} = \frac{0.515}{v} \]
Step 3: Condition for just missing.
The balls must miss by at least their radius sum = 0.05 m (diameter). So: \[ d \geq 0.05 \] \[ \frac{0.515}{v} = 0.05 \Rightarrow v = \frac{0.515}{0.05} = 10.30 \, m/s \] Final Answer: \[ \boxed{10.30 \, m/s} \]