In general, the equations of physics are relationships involving rates of change. This means that given the position, velocity, etc., at one time, we can predict the position and velocity at a later time. As a consequence, it is also necessary to consider the initial conditions of the motion: the initial position and velocity of the object. We will typically describe motion relative to an initial time t = 0, and use the subscript 0 in denoting the initial position x0 (or y0) and velocity v0.
The actual motions of a real, extended object, are far too complicated for us to treat mathematically at this stage. Consider the following scenario: you are cleaning out the gutters on the roof of your home, and slip, falling to the ground. If this were to happen in real life, a combination of panic and instinct would cause you to flail your limbs, hopefully in an attempt to minimize the impact. Because your bones are essentially rigid, we could simplify the motion of your fall by considering only the motions of your head and major joints: the motion of your neck and hips should adequately describe the motion of your torso; the motions of your elbows and wrists should adequately describe the motion of your arms, and the motions of your knees and ankles should adequately describe the motion of your legs:
We can envision, then, using 11 position variables to describe the locations of the various parts of your body as you fell. In addition, although your fall is primarily along a vertical axis, your limbs are free to move in any direction in the plane perpendicular to that axis, as well as along that axis. Therefore each of the 11 position variables must have three parts: an x, a y and a z coordinate, corresponding to the three dimensional space in which you are falling:
Since we can treat the x, y and z coordinates independently, this means we need 33 position variables to describe your fall. Since each part can move independently, we also need 33 velocity variables to describe your fall. Fortunately the acceleration is a constant, for reasons which we will discuss later. But this still leaves us with 66 degrees of freedom (free variables). So including the time, this means that 67 variables and 66 initial conditions are required for our so-called simplification of this problem.
This is clearly unreasonable. We would like to analyze your fall using a single position variable and a single velocity variable, in addition to the acceleration and the time. This will require two major simplifications: rather than describe the position of all of the major parts of your body, we must describe your entire body as a single point. This is called the point particle approximation: we pretend that we have compacted your entire body into a single point located at your center of gravity (a sort of weight-averaged position which we will define more carefully later). This is actually very reasonable because the motion described by the point particle approximation is the same as the motion of your center of gravity in real life. In addition, we will pretend that your body fell in a straight vertical line. This allows us to ignore the non-vertical components of your position and velocity:
With this simplification to two degrees of freedom, we can describe the motion of your fall using four variables and two initial conditions:
y0 is your position at time t = 0 and
y is your position at time t;
We must also be careful to precisely define the origin of our coordinate system: y = 0 at ground level.
v is your velocity at time t;
t is of course the time;
a is the constant acceleration;
v0 is your velocity at time t = 0.
Since the problem specifically states that you fell and were not thrown, v0 = 0. Of course, your velocity changes constantly under the influence of gravity. Since we have defined the origin of the vertical axis to be at ground level, the value of your position decreases from y0 to 0, and therefore your velocity becomes more and more negative (as your speed becomes more and more positive). Since acceleration is the rate of change of velocity, the acceleration must be negative. As we will learn later, the acceleration due to gravity at the earth's surface can be assumed constant; that acceleration is denoted by the letter g and has a value of approximately 9.8 m/s2. So in this problem, as in any problem involving a freely falling object (and ignoring wind resistance), a = - g.
In the problems we will work, the height from which you fell will be given. Since we will know two of the four variables (final height and acceleration) and both initial conditions, we only need two equations in order to completely describe your fall: one for y as a function of t, and one for v as a function of t. We can construct these equations using our knowledge of the units used to measure position, velocity, acceleration and time. The technique we are about to describe is one of the most important things you will learn in this text; you will be using it throughout your study of physics to construct the equations you need to solve problems. Relatively few equations will appear explicitly in this text: this technique will allow you to create them for yourself, using your knowledge of units.
The main idea is this: you cannot add or subtract quantities which do not have the same units. If you walk one half of a mile and run for three minutes, it makes no sense to describe your position with the number 3.5. You would first have to specify the speed at which you ran as, for instance, .1 miles per minute, and then multiply that speed times the three minutes to get .3 miles. Your final position is then .8 miles from where you started. You had to add two numbers which both had the same units before your final result made sense. We will use this idea to construct an equation for position as a function of initial position, initial velocity, acceleration and time.
The first step is to be clear about the goal of the equation you are trying to create: in this case, we want an equation for position, so the left hand side of the equation will be x. Since position is measured in units of distance, the right hand side of the equation will be a sum of terms, all of which are measured in the same units. It is usually helpful to be concrete about the units we are using, so we will use S.I. units of meters and seconds in order to construct our equation. Since initial position is also measured in meters, it is one of the terms on the right hand side of the equation. We would also expect the right hand side to include terms involving the initial velocity and the acceleration. Velocity is measured in units of meters per second, so it must be multiplied by a variable which has units of seconds so that the entire term has the desired units of meters:
v0 t.
Acceleration has units of meters per second squared, so we might expect to multiply acceleration by the square of time in order to get the necessary units of meters. This is almost, but not quite, correct: there is a numerical factor in the acceleration term which has no units, and therefore cannot be predicted by consideration of units alone. Calculus is required to properly derive this unitless numerical factor, although careful experimentation can lead us in the correct direction: if we carefully measure both the position and velocity of a falling object as a function of time, we would find that the constant of proportionality between position and time squared is precisely half the constant of proportionality between velocity and time. But we need calculus to know that this factor of 1/2 is always associated with a squared variable. So the final term is
a t2 / 2,and our general equation for position as a function of time is:
x(t) = x0 + v0 t + a t2 / 2.Note that every term in the equation has units of meters. If you have experience balancing chemical equations, you can think of this as a sort of "unit stoichiometry": instead of balancing the number of atoms of a given element on both sides of the reaction, you are balancing the units in every term of the equation by multiplying factors such that the product of their units gives the desired unit.
There is something else important to notice about that equation: of the six symbols in it (x, x0, v0, t, a and 2), only one is a number: the rest are variables, or at least, parameters (for our purposes, a parameter is a variable which will not be solved for in the equation, like x0, v0 and a in this one). Your mathematical training has taught you that every equation has at most two variables: an independent and a dependent variable, and that a system of n equations must have n variables. Most of the equations you will use in physics have many more variables and parameters than numbers, but in any system of equations, you will always be given enough numerical values for variables and parameters for the system to have a unique solution.
We will often refer to the distance, or change in position x - x0. This is called "delta x" and is denoted using the capital Greek letter delta as Δx:
We still need an equation for velocity as a function of time, but you will construct that equation for yourself. The following applet will not only help you to do that, but will also give you practice in constructing equations based on the units. You will be given the variable on the left hand side of the equation, the variables which will appear on the right hand side of the equation, and any numerical factors which you may need. Your job will be to construct the correct equation by specifying the exponent of each factor on the right hand side. The "Choose factor" button will cycle through all of the possible factors, and the scrollbar marked "Exponent" allows you to chose the exponent for that factor. At any time you can change the exponent on a factor by reselecting the factor and changing the exponent. You must use your knowledge of previous equations to decide on the exponent for numerical factors like 2 or π. In this applet, the velocity or acceleration are sometimes assumed to be zero, so the equations you create will not necessarily be the most general ones possible. But you can put together more general equations (like the one we obtained for position above) by adding together right-hand-sides with the same units.
We said that we need calculus to get the correct coefficient of one half in the term
With a little imagination, you can see that if the intervals got so small that you couldn't see their width, the total
distance traveled would just be the area under the velocity line. So let's use that area to compute the total distance
as a function of time. Since v = t, at any given time the area under the graph is just half the area of a square of side
t. But that gives us
What we have done is to compute a Riemann Sum, taking the limit Δt -> 0. In effect, we
have computed the integral of t*dt. In a calculus class, you would prove that the integral of
The next section continues with an analysis of your fall in linear motion.
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Δx = v0 t + a t2 / 2.
We will also refer to the change in velocity:
Δv = v - v0.
a t2 / 2.
But we can borrow some ideas from calculus to see that it must be true
for constant acceleration. In that case, the graph of your velocity as a function of time is a straight line. To
be specific, we will assume that
a = 1 m/s2
and that you started from rest, so that the equation describing your velocity is
v = t.
Consider how you would figure out the total distance traveled as a function of time. You might
split the time up into intervals, and use the average velocity in the middle of each interval to compute the distance traveled
during that interval with the equation
Δx = v Δt.
Then you could add all of the distances up. But you notice that the smaller the interval, the more accurate the result:
x = t2 / 2.
So that's where the factor of one half comes from.
a xn dx
is
a xn + 1 / (n+1).
So we can see that the factor of one half only arises when n is one; that is, when the acceleration is assumed to be constant,
and there are no higher powers of t in the computation of x.