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 \(y\)-axis at a constant velocity of \(\vec v = v_0 \vec e_y\) to Bob. Bob can catch this ball after the flight time \(t_0 = T/4,\) 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) \[\overline{v'} = \frac{1}{t_0} \int_0^{t_0} |\vec v\,'| dt \] does the ball have from Alice's perspective?

Give the answer in units of \(v_0\) and with an accuracy of 3 decimal places.

\(\)
Hints:

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

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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