What's the big deal with units?

Mistakes when working out problems in physics have two common sources: signs (forgetting a number is negative) and units (e.g. using a number thinking it's in meters when in reality, it's in kilometers).

Watching units avoids a good fractions of the mistakes one can make in solving physics problems. This begs the question: Who cares about solving physics problems? Isn't that something one does only in a physics course? If one gets it wrong, it's a low score. In the grand scheme of things, does it matter?

Consider this. When engineers figure out when to fire the thrusters on a space probe and by how much, that's a physics problem. When a flight crew decide how much fuel their airliner needs to make it to its destination with 200 people on board, that's a physics problem. If they get it wrong, the airplane crashes.

In 1999 a NASA mission, the Mars Climate Orbiter, burned up in the martian atmosphere because the spacecraft received thrust instructions in the wrong units. The system designers, who should have been thinking about units, did not. No-one died, but that was 650 Million dollars gone up in smoke and scientists who never got their data and had to look for something else to do.

In 1983 a jet liner ran out of fuel on its way from Montreal to Edmonton, Canada, before reaching its destination because of a mix up with units. The incident is known as the Gimli glider incident. The only reason the occupants survived is because the pilot happened to be a professional large glider pilot. A large glider is exactly what he had to fly when the engines went silent.

The incident would never have happened if the flight crew had taken a physics course or learned the most important lesson from the one they took: ALWAYS WRITE DOWN YOUR UNITS. EVERY STEP OF THE WAY. They would have spotted their mistake on line 1, when they ended up with a weight in pounds instead of the expected kilograms. Then they probably would not have happily deducted that weight in pounds from a weight in kilograms. And even if they did and carried their units to the last step of the calculations, they would have wondered what an amount of fuel in liter times kilograms per pounds means.

Is Superman gentle enough in catching Lois when she falls off a tall building?

Did you notice how  you had to "give in" when catching a ball to reduce the force from the impact? Giving in helps to give more time to stop the ball. That is really important because the change in momentum is equal to the force applied to the ball (the ball will give you that force right back so you'll feel it) times the amount of time over which the force is applied. To stop the ball, you need to remove all of its momentum. The longer you apply the force, the more momentum you will remove. If you only apply the force for a tiny amount of time, then you must compensate by giving it a high amount of force, and you'll feel it.

When Superman catches Lois falling off a tall building, he must apply an upward force on her to stop her fall. If he applies the force all at once to stop Lois instantaneously, then the force must be huge (because the amount of time is tiny and does nothing to help reducing the momentum). In that case, it doesn't make a difference whether it is Superman or the sidewalk applying that force to stop Lois' fall. Lois will get seriously hurt.

Now for the math:

(time force is applied) x (amount of force) = final momentum (zero) - initial momentum

We can estimate the time the force is applied from the film's footage. Try it below. Count how long it takes from when Superman first makes contact with Lois to when he has stopped her and is starting the ascent back up.

By my estimate, that is 2 seconds.

Lois had time to reach terminal velocity before Superman arrived. The terminal velocity for a human is around 56 m/s. The momentum is the velocity times the mass of an object (or person). Lois looks like she may be around 120 pounds. That corresponds to 54 kg. So Lois has a momentum of about 3000 kg m/s when Superman catches her.

The amount of force applied over 2 seconds by Superman to stop her is 3000/2 = 1500 Newtons. To put that in perspective, that is about 3 times Lois' weight in Newtons. To Lois, it feels like she is taking in 3 g's. Nothing that will hurt or kill her, but definitely something she will feel.

Why do thunderbolts zigzag?

Lightning bolts strike the ground at right angles and they also leave the clouds at right angle because both are conductors (in that context). If the cloud base is not quite horizontal, the lightning bolt will not be straight down. Between the base of the cloud and the ground, lightning then follows a path where there are more impurities in the air. These impurities make it easier to create the ionized track that lightning likes to follow. As you might suspect, the impurities are not organized in straight vertical lines. This is why lightning follows these jagged paths.

For a lot more on lightning, see one of my favorites websites, How Stuff Works. 

To see John Travolta getting zapped, try this simulation (it's related). 

Why is it easier to float in salt water?

The short answer is "because salt water is denser than fresh water". That begs the question "so how does that help floating?"

To answer that question, we need to talk about Archimede's Principle. Archimede's Principle determines the strength of buoyancy. When you float in water, there are two main forces at work: gravity pulling you downward and buoyancy, holding you up so you don't sink. To be able to float, the buoyancy has to counteract the gravity. And in salt water, buoyancy is stronger.

Archimede's principle states that the strength of buoyancy is equal to the weight of the water you displaced by being in it. Consider a boat floating on water (forgive my art):

The boat has displaced a certain amount of water. The weight of that volume of water is equal to the buoyancy force that allows the boat to float. The more water the boat displaces, the higher the buoyancy force, because a larger volume of water weighs more. The boat will sink until there is enough of it under water to increase the buoyancy to a level where it can support the boat against gravity.

This, by the way is why boats need large empty spaces in order to be able to float. You want the boat to be able to displace a lot of water, while not being too heavy. Otherwise, even a large buoyancy becomes insufficient to counteract the downward pull of gravity.

Back to our question. How does the higher density of salt water help with the whole floating deal? In Archimede's Principle, what sets the strength of the buoyancy is the weight of the displaced water. Weight can be increased in two ways: 1) having a bigger volume, as discussed above, or 2) being denser. If the water is dense, then a given volume of it will be heavier and therefore, the buoyancy force will be larger. This is why it is easier to float in salt water than in fresh water.

Quiz: Is it easier to float in oil or in water?

Dad, I'm hungry! Why is the spaghetti taking so long to cook?

Growing up in Switzerland, pasta came with a special set of instructions on the box for cooking at higher elevation.

Pasta is cooked by immersing it in boiling water for some amount of time. At sea level, the boiling point of water is 100 degrees Celsius. At higher elevations, the boiling point is a few degrees lower. With decreasing atmospheric pressure, it is easier for the water to evaporate and it requires less energy (lower heat).

At 1000 meters above sea level, where a good number of Swiss towns are found, the boiling point is down to 96 degrees Celsius. At 2000 meters, were the lower ski resorts are, the boiling point is 93 degrees. And at 3000 meters, were the top of the slopes and mountain cabins are found, the boiling point is down to 90 degrees.

If you are hiking and you are attempting to cook pasta, instead of immersing it in water that is at 100 degrees Celsius, you are immersing it in water that is 10 degrees cooler. You will have to compensate by allowing the pasta to cook for longer.

Why do dogs (and cats) pant when they are hot?

Cats and dogs can't sweat like we do to keep cool. Sweating is an effective way of cooling down the body. It utilizes the difference in thermal energy contained in a liquid versus that contained in a gas. A gas contains more thermal energy (because the gas molecules move fast) than a liquid. Therefore turning a liquid into a gas requires an input of energy.

When it's hot, our body wants to expend some thermal energy so it can cool to a comfortable temperature. When sweating, our body takes some of our water and puts it out on the skin, where it comes in contact with air. In air, the sweat evaporates. The water turns into water vapor, from its liquid form to its gas form. This requires an input of thermal energy, which our body is happy to provide. The water vapor floats away, carrying that extra energy with it and helping us keep cool.

The principle for cats and dogs is exactly the same, except instead of turning sweat into gas, they turn their drool into gas. Like us, they get dehydrated when hot, and so must be provided with plenty of water. Dogs also tend to overheat more easily than cats. When a cat pants, it's really hot.

Why is it deadly to hit concrete when falling from a high place?

This one goes back to a central concept in physics called the impulse-momentum theorem. The impulse momentum-theorem relates a change in velocity with the force that causes this change in velocity.

Warning: What follows is a little graphic.

Say suicidal Johnny leaps off a tall building. On his way down, Johnny speeds up and gains a lot of momentum. When he hits the ground below, he stops suddenly. His momentum goes from a large amount to zero. This happens because the ground exerts a large impulse on Johnny. That impulse is equal to the force the ground exerts on Johnny times the amount of time it takes for the ground to stop Johnny. The latter is very short, as unlike a mattress, concrete is hard. Johnny stops over the course of a few milliseconds. The force from the ground has to be very large in order to compensate for the tiny time interval and still provide the impulse needed to stop Johnny's fall. The force is so large that it causes all kind of damage to suicidal Johnny's body.

Related post: How do ducklings get away with falling from great heights without getting hurt?