Tigers pitcher Doug Fister did not seem to be seriously injured after this line-drive baseball hit to the head. So, that's good. Also, this seems like the video would allow us to answer some questions. In particular, how hard did the ball hit his head? Usually, this is the question most people want to know. They want a value for the force of impact or something like that. This is almost always impossible to determine since it depends on so many different things. You can't just look at the momentum or kinetic energy of the baseball. Neither of these quantities really tell the whole story.
If you want more details, or maybe some gif versions of the hit, check out The Big Lead blog.
How Fast Was the Ball?
This isn't as simple as I would like it to be. Both videos of the ball after the hit show the ball going towards or away from the camera. This makes it rather difficult to get a position-time graph due to perspective issues. However, I can look at the time the ball is in the air along with the distance. According to Wikipedia, the distance from home plate to the pitcher's mound is 18.39 meters. Since Fister was forward of the mound a little bit, let me put the distance at about 17.75 meters (just a guess). From the video, the time it takes the ball to go from the bat to the head is 0.4 seconds (I don't have the best video to work with). This gives an average velocity of:
If you convert 44.4 m/s into mph, you get an average ball speed of 99 mph. Remember, this is just an estimate - is just from 10 frames in the video I used. If I was off by even one frame in guessing when the ball was hit, I could get a speed anywhere from 110 mph to 90 mph. Either way, this is still fast.
What about after the ball hit Doug Fister's head? If you watch carefully, the ball seems to go almost straight up. From the video, I get that it was in the air for 2.8 seconds. If I assume that it is going slow enough to ignore air resistance, then I can calculate the starting velocity. Technically, this would only give me a value for the initial vertical velocity, but I will proceed anyway.
Without air resistance, the ball has a constant vertical acceleration of -9.8 m/s2 and the time to get to the highest point would be half the total time (assuming the ball starts and ends at the same level - which it doesn't actually do). Since the vertical velocity at the highest point is zero m/s, I can use the definition of average vertical acceleration to find the starting speed.
That is about 30.7 mph - in the vertical direction. Let me go ahead and get the horizontal speed also. In this case, the ball landed maybe 30 meters behind the pitcher. If this is true, then the horizontal velocity would just be the distance over the total time giving a value of (30 m)/(2.8 s) = 10.7 m/s or 24 mph.
How Hard Was the Hit?
Like I said, this is what everyone wants. Unfortunately, I can still only guess. In order to get the force the baseball exerts on the head, I need two things. I need the vector change in momentum (which I seem to have a pretty good estimate for) and I need the time of contact between the ball and the head. Even if I did have the time of contact, all I will get is the average force the ball exerts on the head.
Here is where I am going to go. If we think about the forces on the ball during the collision, it is mostly just due to the head pushing on it. The ball pushing on the head and the head pushing on the ball are the same interaction and thus have the same magnitude. With this, I can use the momentum principle which states:
Here is a diagram showing the ball before and after the collision.
Let me assume that the ball is initially only moving in the horizontal direction (which I will call the x-direction). This means I can write the following two expressions for the x- and y-components of the force the head exerts on the ball. What about the gravitational force? I am assuming the force the head exerts on the ball is much larger than the gravitational force so that it can be ignored.
A quick note on notation. I think I might have done something confusing. When I was calculating the velocity of the ball after it hit Fister, I called the initial velocity v1. Now I am using the "1" to represent velocities BEFORE it hit Fister and the "2" for velocities after. Sorry about the confusion.
With this expression, really the only thing I need to estimate is the time of impact. I already have estimates for the above velocities and I know the mass of a baseball is about 0.145 kg. I could guess some values for the impact time (which I will) and that would give the components of the force on the ball (and thus the force on the head). The total magnitude of this force would then be:
Now, what about the time? The impact seems to take less than 1 frame of the video (which is 30 frames per second). This would put an upper limit on the collision time at 0.033 seconds. How about I just completely guess that maybe the lower limit for impact time is 0.001 seconds? Here is a plot of the net force on the ball (and thus the head) as a function of impact times.
Smaller impact times lead to much larger average impact forces. With this estimation, I get an impact force anywhere from 6500 Newtons to 200 Newtons. But what does that mean? Sadly, not much. Really what people want is the damage the ball does. Can a 200 Newton impact do damage? Sure. Could a 6500 Newton impact be mostly harmless? Yes. Why? Well, these impacts depend on so many things. What part of the head did the ball hit? What was the contact area? What was the peak force? How much can the flesh and bone flex before causing damage?
Unfortunately, this isn't a video game where the damage done by some object gives some numerical value to decrease hit points or something. In the end, I still impressed that Fister stayed in the game. He probably had a headache though.
