During a recent Home Run Derby, Aaron Judge did something that no one thought was possible. He took a swing and hit a ball so hard that it collided with the ceiling at Marlins Park. The ball hit the ceiling about 170 feet above the ground. The height of the ceiling had been designed by engineers so that balls wouldn't hit it—but clearly, they can.

OK, I don't really want to talk about sports. I want to talk about physics. Just how would you even calculate the height of a baseball's trajectory? I'm not just going to show you how to do it, I'm going to let you do it too.

Force and Momentum

I'm going to start with the most important physics idea needed for the trajectory of a baseball: the momentum principle. This says that the total force on an object is equal to the time rate of change of the momentum. Momentum is the product of mass and velocity; both it and the force are vectors.

If you know the forces on an object, you can find its change in momentum. With the momentum, you get the velocity and then can find the new position. That's basically how it works.

Two Forces on a Baseball

After a baseball is hit by the bat, it only has two forces on it (OK, approximately two forces. The first is the gravitational force, a downward force that depends on the mass of the object and the value of the gravitational field (g = 9.8 N/kg). The second force on the ball is a little more complicated: It's the air resistance force.

Although you don't think about it much, you've felt this air resistance force before. When you stick your hand out of a moving window or when you ride on a bike you can feel the force as you move through the air. One of the simplest models for this force uses the following equation:

That might look complicated, but it's not too bad. The ρ is the density of air (about 1.2 kg/m³ in most cases). The cross sectional area of the object is A and C is the drag coefficient that depends on the shape of the object. Finally, there is the velocity. This model says that as the velocity increases, the air resistance also increases.

But you might notice one little problem with the above expression: It's not a vector. I left that part off for simplicity, but yes—air resistance is a vector. The direction of this force is always in the direction opposite of the velocity vector.

I can find the values of all of these parameters for air drag, and the mass and size of the ball are easily found online. For this calculation, I will use a drag coefficient of 0.3.

Calculating Trajectory

Isn't this a projectile motion problem? Couldn't you just use the kinematic equations to find the range of a ball after it was hit? Actually, no. This isn't projectile motion because we are including the drag force. Projectile motion problems have an object with the only force being the gravitational force—and this would be approximately true for baseballs at low speeds. We are clearly not dealing with low-speed balls.

You can't use the kinematic equations because those assume the acceleration is constant. However, as the ball slows down or changes direction the air resistance force also changes. With this non-constant acceleration, there is really only one option: Create a numerical solution.

In a numerical solution, we essentially cheat. Since the problem is that forces are not constant, we can pretend they are constant if we take just a tiny time interval (say 0.01 seconds). During this short time, the velocity and thus the air resistance won't change too much, so I could use the kinematic equations (for constant acceleration). This constant force approximation works—but it leaves us with another problem. If I want to calculate where the ball is after 1 second, I would need to do this calculation 100 times (100 x 0.01 = 1). And this is where the computer becomes useful (but not required).

If you want to go over the details of creating a numerical calculation, take a look at this post that models the motion of a spring. Otherwise, let's just jump right into the code. Notice that you can indeed change things in the code and run it again—that's the fun part. Just click the "play" to run it and the "pencil" to edit.

This code is written in Python. That means that the number sign (or as my kids call it, the hashtag) at the beginning of line makes it a comment that is ignored by the program. I added a bunch of comments to point out things that you might want to change (like the initial velocity and the launch angle). Go ahead, change something. You won't break it.

Homework

Since I gave you the numerical calculation, I also have to give you homework.

• Find a launch speed and angle that would produce a home run. You will need to find the home run distance for a particular park. Yes, you should probably find a way to include the height of the wall.
• What is the minimum launch speed that would hit the rafters for Marlins Park?
• For a given speed, what angle gives the maximum range? No, it's not 45 degrees—that's only for motion with no air resistance.
• What would happen if you increased the density of air by just a little bit? Does it make a huge difference?
• My calculation uses a drag coefficient of 0.3—but this is just an approximation. In fact, the drag coefficient changes with the velocity of the ball. See if you can modify the code to include a better drag coefficient. This site might be a good place to start to figure out how to change that coefficient.
• What about the Magnus force? This is another force due to the interaction between the air and a spinning object. See if you can add that force to the numerical calculation.

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You don’t want to be among the first human cyborgs. Because doctors won’t be replacing all your limbs with super-strong robotic ones, and they won’t be giving you cameras for eyes. More than likely, they’ll be saving your life by wrapping your heart in a robot.

Today in the journal Science Robotics, researchers introduced a new kind of device to keep a heart pumping: It cradles the organ and uses a probe to anchor to the wall that separates the heart’s lower chambers. The robot can precisely manipulate a particular chamber, and that could lead to devices that let doctors assist a heart in its normal function instead of relying on a transplant. (Another sort of robotic heart announced earlier this year envelops the organ like a sleeve, but this new robot can work on a single diseased chamber.)

These days, doctors keep a heart pumping blood with something called a ventricular assist device. This is a pump external to the body that helps ferry blood around when the heart just can’t manage on its own. Problem is, because blood is flowing through machinery, the patient has to take blood thinners to make sure the works don’t get gummed up. And doctors don’t like putting people on blood thinners if they can avoid it.

This new robot is incorporated right into the heart, and acts to encourage the organ’s normal function. The bit that rests on the heart is a soft robot made of polymers, meaning it’s, well, soft, so it better conforms to the organ and doesn’t irritate the flesh. But it’s also soft in its operation: Instead of using traditional motors that are complicated and bulky, it’s pneumatically activated, which is a gentler way to manipulate the heart.

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The second bit of the robot is a rod that actually enters the heart and anchors to the wall that separates one ventricle from the other, known as a septum. A needle pierces the septum and a delivery shaft opens up an anchor on the other side of the wall like an umbrella. Then an operator places a disk on the other side to complete the anchor.

So in addition to the soft robot on the outside pumping the free wall of the ventricle, the shaft pulls the septum toward the wall, squeezing the ventricle to get blood flowing. Without pulling on the septum, the device wouldn’t really be replicating the beating of a heart. “The septum is very actively engaged in the ventricular contraction,” says study coauthor Nikolay Vasilyev, a scientist in the Department of Cardiac Surgery at Boston Children's Hospital. “When the heart contracts, it's not only the free walls that move. The septum thickens during the contraction and moves inward into the respective ventricle.”

By manipulating both the outside of the organ with a soft robot and tugging on the septum with a rod, this device helps the heart pump blood much more precisely than other devices: The system reads either the electrical signals from the heart or pressure changes within the ventricle to time its movements in concert with the normal operation of the organ.

The researchers have already shown the robot working in a live pig. The next step could be to actually implant the thing in an animal and stitch it up, then watch the robot work over the course of months.

“In terms of technological development, I believe we are almost at the stage where a large company or a pool of investors take this technology to the next level and make a product out of it,” says University of Leeds roboticist Pietro Valdastri, who was not involved in the study. “I frankly hope this is going to happen, as this technology looks pretty ready to me for this type of jump.”

Robots have already stolen our hearts. Now they're keeping them beating, too.

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If your homemade country loaf comes out of the oven shrunken and unfluffy, you may have neglected a central tenet of breadmaking: Hydrate the flour. When you knead the dough, you are massaging moisture into the wheat’s proteins, creating a matrix of gluten that traps gases so the bread can inflate from the inside. “The gluten structure is stretchy but impermeable,” says Nathan Myhrvold, the tech millionaire, chef, and creator of 2011’s six-volume science-of-cooking megawork, Modernist Cuisine. Now, Myhrvold and his team of food scientists and photographers are back with five more volumes, focused exclusively on bread.

The photos in the $625 labor of loaves, ­Modernist Bread, range from cross-sections of rising sourdough to artful, side-lit layers of injera. But so much about baking, as any practitioner worth their pinch of salt will tell you, takes place at the invisible-to-the-eye chemical level, and Myhrvold wanted to expose that hidden process.

For the image above, his crew rinsed a small ball of dough with water to wash away starch granules and water-soluble proteins, leaving behind a blob of near-pure gluten. Then, using a scanning electron microscope borrowed from Myhrvold’s company, Intellectual Ventures, they captured a slice of the gluten magnified to 734X, tweaking the contrast of the black-and-white image to give maximum visibility to the webby network at the heart of every perfectly quenched, ready-to-bake gob of pre-bread. As Myhrvold puts it, “That web makes wheat breads what they are.” Meaning: chewy, delicious, and upsetting to gluten-sensitive stomachs everywhere.

Satellites do an incredible job of mapping algal blooms, the green mats that spread over lakes and oceans during warm, nutrient-rich summers. But the hypnotic, swirling images from space can't tell if toxins are lurking in a carpet of cyanobacteria, threatening the safety of water.

Ecologists and hydrologists can test water's drinkability by boating through the blooms—though collecting samples off the side of a power boat is tricky and inconvenient. So this year, scientists are monitoring Lake Erie with a robot, 18 feet below the water’s surface.

The so-called Environmental Sample Processor, ESPniagara, sits on the floor of Lake Erie’s western basin. It collects algae from the surrounding water, analyzes microcystin (a small, circular liver-toxic protein), and uploads results for researchers at the end of every test. They're watching this toxin closely, because elevated levels of it could swiftly poison the water supply for humans and wildlife in the surrounding area.

A no-frills charm dominates the ESPniagara's aesthetic. “It kind of looks like a trash can,” says Tim Davis, a molecular ecologist at NOAA in Ann Arbor. Tentacles of clear plastic tubing for sample processing swirl around the lab-in-a-can’s lower half, while circuits and wiring snake between the components above. Those electronics and the machine’s batteries—400 D cell batteries power the unit—understandably need some protection to sit at the bottom of the lake. “The metal trashcan is essentially a pressure case that can withstand very, very high pressures, and essentially keeps it dry,” Davis says.

Staying dry isn’t the only requirement for the lab capsule. ESPniagara also needs to stay put, remain upright, and avoid sinking too deeply into the gunky mud. So NOAA recruited applied physicists at the University of Washington to design the 1,000-pound frame encasing the unit. By their calculations, even if Hurricane Sandy-level winds hit Michigan, the water sampling could continue. And at the lake’s surface, a round orange data buoy relays information from its tests via a cellular modem, like the one in your phone.

So far ESPniagara has been testing the water every other day. But as of August 1, with the risk of harmful blooms steadily rising, it began testing on a daily schedule. It pulls lake water in, concentrating algae cells onto a filter. When the filter is clogged with plenty of algae to measure, the biology begins.

While full-scale labs use temperature-controlled water baths, freezers, and centrifuges to run these kinds of experiments, the ESPniagara accomplishes the same tests with a few carefully formulated protocols. Each toxicity measurement happens within a quarter-sized puck that’s about an inch and a half tall.

To measure the algae’s microcystins, it’s important to know that the cyanobacteria hold most of their toxins inside their cells. “So in order to get accurate concentrations, you need to be able to break the cells open,” Davis says. A bit of methanol-Tween-20 (basically dish detergent) does the trick, along with some heat and pressure. And once the cells are cracked open to reveal all the toxins, the ESPniagara dots samples into a four-by-five grid for quantification.

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The toxin detection relies on antibodies that fluoresce when they’ve bound a specific substance. In this case, the antibodies that don’t bind a toxin light up, so brighter dots mean safer water. An internal camera photographs the test array, and at the end of this whole process—it takes roughly four hours—the data buoy sends off that photo. The results end up with collaborators all the way across the country, at the Monterey Bay Aquarium Research Institute servers. Then Davis and his team download them for their own analysis.

Once they've got toxicity data, they combine it with satellite measurements for algal biomass and hydrodynamic models of windspeed and current. That full picture tells them how toxic the bloom is, and where the toxins will end up next. Knowing that strong winds are about to send more toxins into the water supply, for example, helps treatment plants decide how to act. When more microcystins arrive, they’ll know to roll out extra filtration steps—like particle activated charcoal neutralization—to keep drinking water safe. It’s almost like the ESPniagara gives water treatment plants … extrasensory perception.

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The opioid epidemic is ripping through America like a fire untamed. Blame big pharma, if you want. Blame cheap pain pills and cheaper heroin. Blame the mesolimbic reward system. Just don't wallow in it—the blame. Wallowing takes time, and with opioid abuse killing close to 100 Americans a day, time is in exceedingly short supply. “The number one question is, how do we get a better sense of what's going on in our communities in real-time," says Jeff Beeson, deputy director of the Washington/Baltimore High Intensity Drug Trafficking Area. Not a year from now. Not a month from now. Today.

So last year, the Washington/Baltimore HIDTA set out to create a tool that would give law enforcement and health officials the data they need to respond to the public health crisis as swiftly as possible. The result was a web app called ODMAP that combines street-level data with tools from Esri, the digital mapping company, to help public health officials, police departments, and first responders track and respond to overdoses in real time.

ODMAP's national scope distinguishes it from similar opioid-tracking apps. States across the US are racing to develop tools for managing the country's opioid crisis, and Indiana in particular has successfully tracked trends, collecting data from local agencies to build a statewide database of things like drug arrests, seizures, and administrations of the overdose-reversing drug naloxone. ODMAP takes a similar approach, but focuses on mapping overdoses, specifically—whether they're happening locally, two towns over, or several states away.

That's the kind of geospatial data that can help communities brace for overdoses before they happen. "You've seen those epidemiology maps where a disease spreads outward from an initial set of dots? We're seeing similar patterns on a daily basis with ODMAP," says Beeson. If ODMAP registers a spike in overdoses in Anne Arundel County, Maryland, the app immediately notifies public health officials in Berkeley County, West Virginia, 120 miles away. Why? Because in 8 to 10 hours, West Virginia's eastern panhandle is going to start seeing overdoses, too.

"A lot of these geographic correlations, we didn't know they existed until we started tracking overdoses with the app," Beeson says. By anticipating a ripple effect of overdoses, regional officials can warn their communities, notify hospitals, and ensure first responders have the naloxone they need to administer to overdose victims.

Meanwhile, the same data helps law enforcement officials double-check theories about the way drugs travel into and out of their jurisdictions. Take the relationship between Berkeley and Anne Arundel. Washington/Baltimore HIDTA had suspected a link for years, Beeson says, based on arrest data. "But if we pick someone up for trafficking in Baltimore, it's not like we know where that person's drugs are going. Now, we're basically tracking the drug. We're able to see it in black and white, as it spreads throughout a region." And the more data ODMAP collects, the more regional relationships health and law enforcement officials can confirm. "We've never had overdose data like this before—and we've never shared it with each other," Beeson says.

Health and law enforcement officials I spoke with about the tool say they like it because it's simple, powerful, and free. First responders at the scene of an overdose (Beeson says there are currently close to 1,000 registered nationwide) log in to ODMAP via a password-protected web portal. The interface lets these so-called Level I users specify whether the overdose was fatal, whether they administered naloxone, and, if they did, how much. That information, along with the time and location of the overdose, goes to ODMAP's central database. For first responders, that's all there is to it; the process takes seconds to complete and requires no personal information about the victim.

Police have additional, password-protected access to a form that lets them enter information like the victim's date of birth and overdose history; witness information; whether the victim overdosed on fentanyl, oxycodone, or some other narcotic; whether any drugs were found at the scene; even a photograph of the drugs' packaging. That information is stored on a separate database, to keep everything HIPAA compliant. If any of the data matches a previous overdose report, ODMAP will connect the reporting officers so they can coordinate.

Things get more interesting on ODMAP's backend, which is accessible to Level II user like sheriffs and public health chiefs. It requires a separate username and password and provides a bird's-eye view of overdose incidents at the national level. Every overdose appears on the map in the form of a color-coded blip. Accessing data for a state, county, or neighborhood is as simple as panning and zooming.

"It's like having real-time traffic data on Google Maps," says Aaron Kustermann, chief of intelligence for Illinois State Police, one of several organizations in the state that recently began using ODMAP. "Being able to monitor trends and see what's happening and compare with other areas? That's where the power is." Understanding where and how people are overdosing can take local health and law enforcement months to suss out. "With this tool, we can react in real time to a spike in fentanyl-related deaths, or purity-related deaths."

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Adoption will make or break this tool. "From a surveillance perspective, the more data you have on a given health issue the better," says Harvard University's John Brownstein, a computational epidemiologist who uses digital tools to map disease and drug abuse across populations. "Having direct data on overdoses is super exciting, but it's a crowdsourcing tool, so you want as much engagement as possible."

From the looks of it, ODMAP is spreading quickly. It launched on January 18th in just two West Virginia counties. Today, 70 counties across 19 states are actively contributing data to the system. Last week there were 16 states on board, and in the past few days, the number of health and law enforcement agencies using the program jumped from 168 to 186. The opioid crisis might be sweeping the country, but so, too, are the tools that could curb its spread.

“Ask any health or law enforcement agency in the country: We don’t have the time, and we don’t have the resources to sufficiently deal with the opioid crisis,” Beeson says. “We can’t throw money at it, and we can’t arrest our way out of it. But what we can do is use data and technology as a tool, to maximize what limited resources we have.”

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