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Chapter 5: Problem 62

A softball, of mass \(m=0.250 \mathrm{~kg}\), is pitched at a speed \(v_{0}=26.4 \mathrm{~m} / \mathrm{s}\). Due to air resistance, by the time it reaches home plate it has slowed by \(10.0 \% .\) The distance between the plate and the pitcher is \(d=15.0 \mathrm{~m}\). Calculate the average force of air resistance, \(F_{\text {air }}\) that is exerted on the ball during its movement from the pitcher to the plate.

Short Answer

Expert verified
Answer: The average force of air resistance on the ball during its travel is approximately \(F_{air} \approx -1.98 \text{ N}\) (negative sign indicates the force is opposing the motion).

Step by step solution

01

Calculate the final speed of the softball

The initial speed, \(v_0\), of the softball is 26.4 m/s and the air resistance reduces its speed by 10%. We can calculate the final speed, \(v_f\) as follows: \(v_f = v_0 - (\text{reduction percentage}) \times v_0\) \(v_f = 26.4 - 0.1 \times 26.4\) \(v_f = 26.4 - 2.64 = 23.76 \text{ m/s}\)
02

Calculate initial and final kinetic energies

The initial and final kinetic energies of the softball can be calculated using the following formula: \(KE = \frac{1}{2}mv^2\) Initial kinetic energy, \(KE_i\): \(KE_i = \frac{1}{2} (0.250)(26.4)^2\) Final kinetic energy, \(KE_f\): \(KE_f = \frac{1}{2} (0.250)(23.76)^2\)
03

Use the work-energy principle to find the work done by air resistance

According to the work-energy principle, the work done by air resistance is equal to the change in kinetic energy. \(W_{air} = KE_f - KE_i\) Calculate the work done by air resistance: \(W_{air} = \frac{1}{2}(0.250)(23.76)^2 - \frac{1}{2}(0.250)(26.4)^2\)
04

Calculate the average force of air resistance

The force of air resistance can be calculated using the formula for work, \(F_{air}=\frac{W_{air}}{d}\), where \(d\) is the distance between the pitcher and the plate. \(F_{air} = \frac{W_{air}}{15.0 \text{ m}}\) Using the calculated value of \(W_{air}\), calculate the average force of air resistance: \(F_{air} = \frac{ \frac{1}{2}(0.250)(23.76)^2 - \frac{1}{2}(0.250)(26.4)^2 }{15.0}\) Now, simply compute the value for \(F_{air}\).

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