I work very hard to differentiate my physics class from merely an applied mathematics class. We do quantitative demonstrations nearly daily, in which a mathematical prediction is checked via direct measurement. I frequently ask on homework problems, "Justify the physical reasonability of this answer." My class is incessantly discussing how to figure out whether or not answers make physical sense, regardless of whether arithmetic is done correctly or not.

I amended a problem the other night from (I think) the Serway & Vuille text. I gave them the graph to the right, and wrote:

1.A possible force vs. time curve for a ball struck by a bat is shown in the figure.

(a) Calculate the impulse delivered to the ball.

(b) This 0.25 kg ball was initially moving toward the bat at a speed of 20 m/s. Calculate the exit speed of the ball.

For part (a), most of the class figured out to take the area under the graph, which they better have -- that same day in class I had discussed how impulse can be found as the area under such a graph. Some students estimated an average force, which would be around 4000 or 5000 N, and multiplied by 1.5 ms. Fair enough.

**Understandable mistake #1:**A few made the error of multiplying the MAXIMUM force of about 8,000 N by the 1.5 ms time interval. I took off one point out of fifteen for that -- these students were at least approaching the problem with relevant physics. This mistake makes the impulse wrong by a factor of 2.

**Understandable mistake #2:**A few also failed to read the horizontal axis, and multiplied by a time interval of 1.5 s, not 1.5 ms. These students also were approaching the problem with correct physics, but made an arithmetic error. Granted, the answer for impulse was off by a factor of 1000, giving them 7500 Ns instead of 7.5 Ns. But I can't really expect anyone to have a serious physical understanding of orders of magnitude for impulse calculations, especially in the first two days of studying the topic. So I took off just one point.

On to part (b).

**Understandable mistake #3:**The most common error was to fail to account for the direction change of the momentum vector. The ball has a momentum of 5.0 Ns toward the bat before the collision. The ball's momentum changes by 7.5 Ns. But, that doesn't give the ball a final momentum of 12.5 Ns! Since the ball changed directions, the momentum must have first DECREASED by 5.0 Ns to zero, and then increased in the direction away from the bat by 2.5 Ns.

With the failure to account for the direction change of the ball, the exit speed works out to 50 m/s -- a lot, but still not unreasonable, as baseballs hit for home runs routinely exit the bat with speeds above 100 mph. The correct answer is 10 m/s, or about 22 mph -- not a very hard hit, but also not unreasonable. Anything related to a baseball in the tens of miles per hour is just dandy. I took off just two points for failure to account for direction in a momentum calculation.

(Aside -- I'm an American, and I watch baseball. However, if someone had assumed no direction change for the ball, given me the 50 m/s answer, and then discussed how the cricket batsman is allowed to propel the ball in the direction in which the ball was already moving, that student would have earned full credit and a piece of candy.)

What about the student who compounded error upon error?

**Understandable mistake, combo platter:**The student who used the max force rather than the average force to calculate impulse, AND who didn't account for the ball's changing direction, got something like 68 m/s. Well, that's about 150 mph, and still not horrid -- after all, that's only 40% above the typical 100-120 mph exit speeds in the majors. (I'm recalling my

*Physics of Baseball*by Robert Adair; I hope I have that value right. Please correct me in the comments if I'm wrong.) That answer loses only the two points for the direction change issue.

**Terrible, Horrible, No Good, Very Bad Mistake:**Oy. I had several students who failed to see the units of ms, and found an impulse of 7500 Ns. Then they carried through the mathematics in part (b) to find an exit speed of 30,000 m/s. Oh, I say, whoa there. Really? 30 km/s? Mach 88? Okay, Bugs Bunny threw faster than that in the Christopher Columbus episode (when he threw a ball around the world in about 2 or 3 s)... but other than in a cartoon world, NO. These students lost an ADDITIONAL 5 points out of 15. One sheepishly said, "well, the math said 30,000 m/s, but you didn't ask us to justify the reasonability." I pointed out that in physics, physical reality always will trump mathematical manipulation. It doesn't matter whether I *ask*, one should *always* be conscious of physical reasonability.

**Epilogue:**One lone student made the Terrible Horrible No Good Very Bad Mistake, got 30,000 m/s... and pointed out "that is nonsense, a baseball can't ever go faster than 100 m/s or so, the answer is ridiculous but I don't know what I'm doing wrong." He lost the one point for failure to read the graph properly, and one point for an incorrect answer... and that was all. Reward those who demonstrate their commitment to physical reality.

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