Playground physics

Alice and Bob are having fun throwing a ball to each other on a merry-go-round. Charlie looks at the game from outside of the merry-go-round. From his perspective, the ball thrown by Alice flies straight along the yy-axis at a constant velocity of v=v0ey\vec v = v_0 \vec e_y to Bob. Bob can catch this ball after the flight time t0=T/4,t_0 = T/4, because the merry-go-round has completed a quarter turn in the meantime.

But how does Alice observe (rotating reference system)? What average speed (average of absolute value of velocity vector) v=1t00t0vdt\overline{v'} = \frac{1}{t_0} \int_0^{t_0} |\vec v\,'| dt does the ball have from Alice's perspective?

Give the answer in units of v0v_0 and with an accuracy of 3 decimal places.


Hints:

  • The merry-go-round rotates with constant frequency ω=2π/T\omega = 2 \pi/T, so that Alice ((point rA)r_A) and Bob ((point rB)r_B) move on circular paths in the stationary reference system. In the rotating reference system, both points rAr_A^\prime and rBr_B^\prime are stationary.
  • Search for a 2×22\times 2 matrix D\mathbf{D} such that the transformation reads r=Dr\vec{r}^\prime = \mathbf{D} \cdot \vec{r}. For the calculation of the average velocity, you may use the integral: 1+x2dx=12(x1+x2+arcsinh(x)). \int \sqrt{1 + x^2} d x = \frac{1}{2} \left( x \sqrt{1 + x^2} + \text{arcsinh}(x) \right).

Bonus question: Which (fictitious) forces act on the ball in the rotating reference system? How can we explain the path of the ball?

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