Problem

A bat strikes a 0.145 kg baseball. Just before im…

05:33

Problem 43 Easy Difficulty

. Bone fracture. Experimental tests have shown that bone will rupture if it is subjected to a force density of $1.0 \times$ $10^{8} \mathrm{N} / \mathrm{m}^{2} .$ Suppose a 70.0 $\mathrm{kg}$ person carelessly roller-skates into an overhead metal beam that hits his forehead and completely stops his forward motion. If the area of contact with the person's forehead is $1.5 \mathrm{cm}^{2},$ what is the greatest speed with which he can hit the wall without breaking any bone if hiss head is in contact with the beam for 10.0 $\mathrm{ms}$ ?

Answer

2.14 $\mathrm{m} / \mathrm{s}$

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Physics 101 Mechanics

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Video Transcript

in this problem. We're told that the maximum amount of force per area that a bone can endear before breaking is tense the eight units per square meter. So we're asked, if a 70 kilogram person with a forehead area of 1.5 times 10 to the negative four square meters or 1.5 square centimeters hits a bar, Um, and spend 0.1 seconds in contact with it. What's the maximum speed with which they can hit a bar before breaking a bone? So, um, we first want to calculate the maximum force that they can endure. Um, and from that we can get the maximum impulse over 0.1 seconds end from that. Since impulse is the change in momentum, we can easily get the initial speed that would produce that impulse. So, um, let's find theme maximum impulse first. So, um, imports is equal to force times the time over which it's applied. Now, um, let's say that the direction of the person's initial motion is positive. Then, since Thea Bar that the person collides with is pushing back against them, the force will be negative. So it's going to be negative f max over area which were given right here, Um, times the area of their head ands that is multiplied by Delta t. So we have all these constants. It's negative one times 10 to the eighth you didn't per square meter times 1.5 times tends the negative four square meters times 0.1 seconds. And plugging this into a calculator, we finds that the maximum impulse is negative. 150. You didn seconds. Now we also know that impulse is the change in momentum. So it's equal to Delta P. And we also know we can just substitute in, um, the mass times final, uh, velocity, which we know is zero minus the initial velocity, which we're calling v Max, since it's the maximum before breaking a bone that's equal to negative 150 nude in seconds. So V f is zero. Which means that, um, isolating for v max so multiplying both sides by negative one and dividing both sides by mass. The max is equal to 150 nude in seconds over the mass, which is 70 kilograms, and we just need to plug this into a calculator. Defines that the maximum velocity is 2.1 meters per second

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