As Felix Baumgartner prepares himself to jump from the edge of the atmosphere, we can reflect on a couple of Physics ideas. One has to do with the laws of motion as put forward by the great Sir Isaac Newton and the other is about the general confusion between accuracy and precision. Before we go any further, let me just say that, in all instances, I’d rather be accurate than precise.
So the premise is this: Felix will go up in a balloon to an altitude of about 36.5km and then jump. He will have a parachute with him (hopefully in working order) and this will only be deployed when he is much, much closer to the ground. Before then, for the initial part of the fall, he will be in free fall until he hits terminal velocity and fall at a constant speed towards the Earth. The aim, I take it, is to be able to reach and exceed the speed of sound in air. That is, Felix is trying to fall at a speed of at least 330 metres per second. A daring feat indeed!
As always, there is a lot of Physics going on here. We’ve seen that the principles of Physics permeates all aspects of life, from the microscopic to the astronomically big, for things which occur in nanoseconds to events that last billions of years, from what we can relate to directly (such as gravity) to the esoteric (such as the Higgs boson). The challenge that Felix is about to undertake is not one to be left to chance. Some experiments can be carried out in a trial and error way and we then try and find some pattern with the outcomes of the experiments. In Mr Baumgartner’s case, he cannot subject himself to trial and error and see which method of jumping and falling and landing is safest for him! A lot of preparation has to go into finding out how to make this task safe yet daring enough to be, shall we say, awesome. Calculating the altitude at which he should deploy his parachute is one of them, calculating if whether he will actually be able to travel faster than sound is another and calculating how to land safely is yet another. Of course, in all of these calculations both accuracy and precision are important.
Before Felix steps off from the ledge of the balloon, he will not fall to the ground. That sounds obvious, right? Before he steps off the ledge, he won’t fall – obvious! But why? Why should that be obvious? Why isn’t something going to happen before it happens? Well, one way to start thinking about this is to see what is stopping Felix from falling. Again, that is obvious. The change from not falling to falling is Felix stepping off the ledge. And you can say that once he’s off the ledge then gravity is what makes him fall. We all know that! Gravity makes things fall. But it’s not like gravity kicks in as soon as he steps off the ledge and pulls him to the ground. Gravity is always acting on Felix regardless of whether he is on or off the ledge. How can we test that? If he stood on a bathroom scale while on the ledge, it will show his weight. But wait! What is thing called weight? It is nothing more than a good example of Newton’s second and third law of motion.
What Newton’s second law tells us, in simple terms, is not F = ma. That’s the most common misconception about the second law. Yes, F = ma is a valid equation, yes it comes from Newton but it is only a special case of the second law. What the second law tells us is this: the rate of change of a body’s momentum is directly proportional to the force acting on the body. By momentum we mean the product of the body’s mass and its velocity. Simple, multiply the mass by the velocity, you get momentum and see how this momentum is changing with time and you’ll be able to deduce what force is acting on that body. And if the momentum isn’t changing then there cannot be a net force acting on the body. A constant momentum most likely means that the velocity of the body isn’t changing. That is, the body is moving at constant speed and in the same direction throughout its motion. (Of course there can be a situation where the mass of the body is changing while its velocity is constant so that the momentum changes as a result. But in Felix’s case, I don’t think his mass will change – at least not perceptively – in order to account for the change in momentum. The only change in momentum in his case will be down to his change in velocity.) So, to summarise this elegantly, we have F = dp/dt where dp/dt is the mathematical way of saying how momentum (p) is changing with time (t). The d in dp and dt stands for “a small change in”. As time goes by, as time changes little by little, momentum also changes, little by little. How much momentum changes (dp) over that change in time (dt) is what gives us this ratio dp/dt. And therefore according to Newton, this ratio has to be directly proportional to (if not equal to) a force (F) acting on that body. The greater the force, the more the momentum will change over time.
So how do we get from F = dp/dt to F = ma? Well, since momentum (p) is the product of a body’s mass (m) and its velocity (v), then we can write this as an equation: p = mv. Any change in momentum, dp, can be expressed as d(mv). Now, because we assume that mass will not change, then the only thing that will change in d(mv) is the velocity. So we can safely assume that dp =m(dv). Dividing both sides by dt we get, dp/dt = m(dv/dt). Now, we have a name for this (dv/dt) term. It’s the rate of change of velocity of a body. It is how fast or how slow the velocity of an object is changing over time. We call this acceleration. The greater the acceleration, the quicker the velocity will change. When sport car makers boast about doing 0 to 60 in 3 seconds, or 0 to 60 in 1 second then all they are bragging about is which one has the bigger acceleration. It’s not only what your top speed can be that matters in a sports car but more importantly it’s how much it can accelerate. And if we denote acceleration by a then we have a = dv/dt. Which brings us back to our expression for rate of change in momentum. We have dp/dt = m(dv/dt) = ma. And, finally, because F = dp/dt, we end up with the famous F = ma. But remember, this is valid only if the mass (m) of the body is not changing. True, in most cases that will be valid but strictly speaking, it is not Newton’s second law of motion.
Armed with our newly derived F = ma, how can we relate this to Felix’s weight? We’re half way there. As mentioned earlier, weight is but an example of the second and third law of motion. We now need the third law to round up what weight stands for. Standing on a surface is, for the least, a banal thing. But what is going on is the action and reaction of forces between our feet and the surface. No matter how repulsive we can be as human beings we are constantly being attracted towards the Earth by the force of gravity. The force of gravity acting on us must therefore be responsible for some sort of acceleration on our body according to F = ma. This acceleration due to the force of gravity is a special type of acceleration called the acceleration due to gravity. I know, it won’t score highly in an appellation contest, but it does what it says on the tin. And we use the letter g to denote this special type of acceleration. From several experiments carried out over centuries (typically involving swinging pendulums in the school labs), we’ve calculated that g is approximately equal to 9.81 m/s2 and it can be assumed to be constant when we are relatively close to the ground. The further away we move from the surface of the Earth, the smaller g becomes. That is to say, the acceleration felt on a body close to the ground is stronger than that felt by a body hovering above the Earth at an altitude of tens of kilometres. How much stronger, well, hardly significant for most practical purposes but stronger nevertheless. While we remain within a few tens of metres from the ground, we can safely assume that g will remain at 9.81 m/s2. Someone with a mass of 70 kg will feel this acceleration acting on her. Someone else with a mass of 50 kg will also feel exactly the same acceleration acting on him as long as they are both at the same height relative to the ground. The value of g is not dependent on the mass of a body. It is a characteristics of the planet Earth and changes only relatively to the Earth. We’ve mentioned that the further away we are from the Earth, the weaker the acceleration becomes. Not only that, but if the mass of the Earth was less than what it actually is (roughly 6 x 1024 kg) then g would be less than 9.81 m/s2. So g is totally dependent on the mass of the Earth as well as the distance we are from the surface of the Earth.
The force, therefore, acting on a mass of 50 kg is simply the product of the mass and the acceleration due to gravity acting on it. This gives us 50 kg multiplied by 9.81 m/s2 which is equal to 490.5 N. Fine, you say, we kind of get it how we come to a value of 490.5 but where does this N come from? N here represents the unit of force and stands for Newton. Yes, the very same guy who stipulated the law of gravity. The force of 490.5 N is what we call the weight of the body. In our day to day lingo we interchange the words ‘mass’ and ‘weight’. I suppose we can live with that and say that a packet of flour weighs 1 kg when what we actually mean is that the packet of flour has a mass of 1 kg. When we’re told at the check-in desk at the airport that our luggage is overweight at 27 kg then we can easily ignore this lapsus linguae. Our luggage is overweight at 265 N or ‘overmass’ at 27 kg. That makes more sense but, again, we are so used to swapping mass for weight that we think they mean the same thing. In Physics, however, mass and weight are different but related. So weight has to do with the force with which the Earth pulls a body towards it. This is where we bring in Newton’s third law of motion.
In plain English it says, for every action there is an equal and opposite reaction. There are a few subtleties to this law but we do not necessarily have to go in such details here. In other words, when we push against something (a wall for example) then there is a force of equal magnitude pushing against us (the wall pushes back). The ‘action’ is acting on the wall while the ‘reaction’ acts on us. Not all pair of forces are action-reaction pairs. But when we push against the floor with our feet or, to put it another way, when we stand, the floor pushes back against our feet with the same force (same in magnitude but opposite in direction). One force is acting on the floor, the other on our feet. It is this play of opposing forces that gives rise to our sensation of weight. Had there not been a floor to push against us, we would not have felt that force acting on us and we would therefore not have been aware of our weight. This notion of weight is very much dependent on the fact that we can feel some sort of resistance against us pushing down towards the ground. When we fall, it is obvious that there is nothing opposing our downwards motion yet it does not mean that there is no force acting on us. The Earth is constantly pulling us, attracting us towards it. Why we fall in some cases and not in others is because sometimes we have a surface stopping us from falling (a floor or a ledge for example) and sometimes we have nothing to prevent us or hold us back and we fall towards the ground. Imagine standing on a table and by some mysterious means the table were to disappear in thin air leaving us at about a metre above the ground. Then, all of a sudden, we would fall to the ground. But whether we are standing on the table or falling towards the ground, the Earth is constantly pulling on us. When we are on the table, the table is pushing back on us and we feel that force as our weight. When we’re falling we don’t feel that opposing force but it doesn’t mean it’s not there. So for that fraction of a second, we are weightless even though gravity is still pulling us down. It’s only when we reach the ground that we feel that opposing force called weight. Every object which is in free fall is weightless. By free fall we mean that the only force which is acting on the falling object is the force of gravity. There is no friction or air resistance or drag or anything of that sort. Only gravity.
When Felix will step off the ledge, he will be weightless but he won’t remain at an altitude of 36.5 km hovering over our heads because the Earth, as always, will keep pulling him down. And so he will fall. Why will he not remain at this height? Why does he have to fall at all? Yes the Earth is pulling him with the force of gravity. But so what? Well, we have to call upon our old friend, Isaac, once again. He also came up with the first law of motion. And this says, I won’t budge if you don’t nudge. Well not in those exact words but the idea is that, a body will not move unless a force is acting on it. And not only that, but if something is already moving then it will keep moving unless a force causes it to stop. This is the most basic law of motion. To change the state of rest or motion of a body one must apply a force. The converse is also true in that, if there is a force acting on a body then it will change its state. That bottle of water on my desk will remain motionless for hours, days, years, until the end of time if no net force acts on it. And if I observe it moving then I can immediately deduce that there must be a net force acting on it. So Felix, by being off the ledge will not feel any opposing force on him and therefore the net force acting on him will be the force of gravity. With the force of gravity acting on Felix, he has no choice but to follow what Isaac said and move. Well, I say move but it’s more like falling, i.e. moving downwards in the direction in which the force is acting. This is why he falls. While he is in free fall (which will only last a very brief moment anyway because air resistance) he will be weightless.
So, in that very specific case of a skydiver attempting to jump from way up high, we have been able to relate where and how Newton’s laws of motion apply. But the lesson to be learnt here is that these laws are not restricted to jumping daredevils only but it affects our every day life in whatever action we do. We push the door, it moves. The harder we push it the faster it moves. We stand or push on the ground, the ground pushes back, we feel our weight. We toss a coin and it rises but gravity gradually wins over that small force we imparted to the coin at the beginning and it makes the coin fall back towards the ground. We push back on the floor with our feet, friction pushes back, and we are therefore able to move forward and walk. Without friction there is nothing to push back and all we do is slip backwards; this is why we skid on a slippery floor. There is no need to become entangled in equations in those situations and start calculating forces and ratios of momentum and time but it is essential that we realise that these seemingly banal actions rest on the laws of motion as stated by Sir Isaac Newton. And it is also thanks to these laws that we’ve been able to understand and master this force of gravity and send astronauts into space and explorers to the moon and spacecrafts to the edge of the Solar System.
Now, to come to our second concept: accuracy and precision. When we say that g near the surface of the Earth is 9.81 m/s2, we have to bear in mind that this is just an approximation of the real value of the acceleration at a certain point. The real value could be 9.80665 m/s2 but by stating it as 9.81 m/s2 we’re being less precise not less accurate. A less accurate answer would be 14.21 m/s2. By giving this number as 14.21325644 we are not being any more accurate just more precise with our (wrong) answer. Accuracy has to do with how close we are to the true value while precision tells me how wide is my margin of error. Having a small margin of error is pointless when the measurement itself is way off the correct value. I’d rather someone tell me to go Paris to see the Eiffel Tower than send me to this very precise GPS coordinate points (-33.856, 151.214). Those coordinates will not send me anywhere near the tower not even if we increase the precision to (-33.856973, 151.214486). So even if the first set of instruction was quite vague, I wasn’t told exactly where in Paris to go to so that I can see the Eiffel Tower, by being in Paris I was closer to my target than by being at those coordinates: (-33.856973, 151.214486). These coordinates, in fact, pinpoint the exact location of the Sydney Opera House. As you can see, it’s more important to be close to the target, the true value, than to have a very precise measurement which is far from the truth. Of course, if you can have both an accurate and precise measurement then perfect. Neither accurate nor precise values will send you right off track. That is why I said at the start that in all instances, I’d rather be accurate than precise. And I hope that when determining the risks of jumping off the ledge at the edge of the atmosphere, Felix and his team are being both accurate and precise in their measurements and calculations. But, more importantly, that they realise that it does not matter how precise you are in your measurements and calculations if you are not accurate.