Question

1. zak throws a fastball at 85 mph and yaser catches it. yaser, as all good baseball catchers do, starts with his gloved hand outstretched towards the pitcher to absorb the impact of the ball. while catching the ball, yaser’s hand moves backward a distance of one foot. that is, the ball goes from 85 mph to rest in a distance of only one foot. the mass of a baseball is 145 grams. calculate the average force yaser exerts on the ball while stopping it. give your answer in units of newtons to two significant digits. for the curious, there are 4.45 newtons in a pound, so you can calculate the force in pounds as well.

Explanation:

Step1: Convert units to SI

First, convert speed from mph to m/s. 1 mph = 0.44704 m/s, so 85 mph = 85 * 0.44704 ≈ 37.9984 m/s. Mass of ball: 145 grams = 0.145 kg. Distance: 1 foot = 0.3048 m.

Step2: Use kinematic equation

We use \(v^2 = u^2 + 2as\) to find acceleration \(a\). Final velocity \(v = 0\), initial velocity \(u = 37.9984\) m/s, \(s = 0.3048\) m. So \(0 = (37.9984)^2 + 2a(0.3048)\). Solving for \(a\): \(a = -\frac{(37.9984)^2}{2*0.3048} ≈ -\frac{1443.88}{0.6096} ≈ -2368.57\) m/s² (negative for deceleration).

Step3: Apply Newton's second law

\(F = ma\), where \(m = 0.145\) kg, \(a = 2368.57\) m/s² (magnitude). So \(F = 0.145 * 2368.57 ≈ 343.44\) N. Rounding to two significant digits: 340 N (or 3.4×10² N). For pounds: 343.44 N / 4.45 N/lb ≈ 77.2 lb ≈ 77 lb.

Answer:

The average force in newtons is approximately \(\boldsymbol{3.4 \times 10^2}\) N (or 340 N), and in pounds is approximately \(\boldsymbol{77}\) lb.