Question

An inexperienced catcher catches a \(130 \mathrm{km} / \mathrm{h}\) fastball of mass \(140 \mathrm{g}\) within \(1 \mathrm{ms}\), whereas an experienced catcher slightly retracts his hand during the catch, extending the stopping time to 10 ms. What are the average forces imparted to the two gloved hands during the catches?

Short answer

Answer: The average forces experienced by the catchers are: - Inexperienced catcher: approximately -5,066 N - Experienced catcher: approximately -506.6 N

Step-by-step solution

List down the given values

We have the following values given in the problem: - Initial velocity of the baseball, \(v_i = 130\, \mathrm{km\,h^{-1}}\) - Mass of the baseball, \(m = 140\, \mathrm{g}\) - Stopping time for inexperienced catcher, \(t_1 = 1\, \mathrm{ms}\) - Stopping time for experienced catcher, \(t_2 = 10\, \mathrm{ms}\)

Convert units to the standard units(SI units)

We need to make sure all values are in the standard units: - Convert \(v_i\) to \(\mathrm{m\,s^{-1}}\): \(v_i = 130\, \mathrm{km\,h^{-1}} \cdot \dfrac{1000\,\mathrm{m}}{1\,\mathrm{km}} \cdot \dfrac{1\,\mathrm{hr}}{3600\,\mathrm{s}} = 36.11\, \mathrm{m\,s^{-1}}\) - Convert \(m\) to \(\mathrm{kg}\): \(m = 140\, \mathrm{g} \cdot \dfrac{1\, \mathrm{kg}}{1000\, \mathrm{g}} = 0.14\, \mathrm{kg}\) - Convert \(t_1\) and \(t_2\) to \(\mathrm{s}\): \(t_1 = 1\, \mathrm{ms} \cdot \dfrac{1\, \mathrm{s}}{1000\, \mathrm{ms}} = 0.001\, \mathrm{s}\) and \(t_2 = 10\, \mathrm{ms} \cdot \dfrac{1\, \mathrm{s}}{1000\, \mathrm{ms}} = 0.01\, \mathrm{s}\)

Calculate the average force for the inexperienced and experienced catchers

Now, we can use the force formula for both catchers, keeping in mind that final velocity, \(v_f = 0\): For inexperienced catcher: \(F_1 = \dfrac{m(v_f - v_i)}{t_1} = \dfrac{0.14\,\mathrm{kg}(0 - 36.11\,\mathrm{m\,s^{-1}})}{0.001\,\mathrm{s}} = -5,066\,\mathrm{N}\) (Negative sign indicates opposite direction) For experienced catcher: \(F_2 = \dfrac{m(v_f - v_i)}{t_2} = \dfrac{0.14\,\mathrm{kg}(0 - 36.11\,\mathrm{m\,s^{-1}})}{0.01\,\mathrm{s}} = -506.6\,\mathrm{N}\)

State the results

The average forces imparted to the gloved hands during the catches are: - Inexperienced catcher: \(F_1 \approx -5,066\, \mathrm{N}\) - Experienced catcher: \(F_2 \approx -506.6\, \mathrm{N}\) Remember, the negative sign indicates that the force acted in the opposite direction to the initial velocity of the baseball.