Kinematics.

One-dimensional (straight line) motion.

Kinematics is the "geometry of motion". It describes motion without using the concepts of force and mass. In this chapter our interest is in Galileo's pioneering work in kinematics, treating the problems of constantly acceleration motion, including the motion of falling bodies.

Definitions:

[1]

Velocity: v = (x₂ − x₁)/(t₂ − t₁), which may be written v = Δx/Δt

[2]

Acceleration: a = (v₂ − v₁)/(t₂ − t₁), which may be written a = Δv/Δt

The symbol Δ is an operator meaning "change in". We will use boldface symbols for position, displacement, velocity and acceleration in our equations to emphasize that these are quantities that can have either positive or negative signs, and that the sign indicates direction of motion along a line. It is very important that you specify the direction you choose to call "+" before beginning a problem, for the relative signs matter. Once you make that choice, do not change it while doing the problem. We are also anticipating the fact that these quantities are vector quantities, and when we extend this presentation to include two and three dimentional motion, we will nead to treat certain physical quantites as vectors, and boldface is the traditional way to indicate that quantities are vectors.

The above equations are valid for any kind of motion if you take the calculus limit of the right sides as Δt goes to zero. But in the special case of straight-line motion where the acceleration is constant, the velocity is the same whether you use a large or small time interval, Δt, and since the acceleration is constant, it also doesn't depend on the size of the time interval. The velocity-time graph is a straight line, and we can write:

The velocity-time relation for constant acceleration. The average velocity over an interval is the mean of the instantaneous velocities at the endpoints.

[3]

Average velocity: <v> = (v₁ + v₂)/2

[4]

And <v> = d/t

These two may be immediately combined to give the equation

[5]

(v₁ + v₂)/2 = d/t

for any two points 1 and 2 on the straight line. d = x₂ − x₁ is the displacement in position that occurs during the time interval, t.

This is the last time we will make use of the concept of average velocity, for it isn't a useful concept in problem-solving. To tell the truth, it is only used here to avoid introducing calculus, but its use is legitimate in this special case of constant acceleration.

Initial conditions assumed: The body has displacement d = 0 at t = 0, and its velocity at t = 0 is v_o. We sometimes refer to t = 0 and t as the "initial" and "final" velocities, but in fact they are simply the times at which the moving body happens to be at two different positions, chosen because they are of particular interest to us in problem-solving. The labels "initial" and "final" refer only to the endpoints of the time interval being considered.

The equations below are results valid only for the case of constant acceleration.

[6]

Velocity: v = d/t

[7]

Acceleration: a = (v − v_o)/t

Use equation 5 written for the "initial" and "final" velocities v_o and v. Then solve the equation for d.

d = (v_o + v)t/2

Solve equation 7 for v

v = v_o + at

Put this into equation 9.

[11]

d = v_ot + (at²/2)

Finally, multiply equation 7 by equation 9, and rearrange to get:

[12]

v² = v_o² + 2 a•d

We have shown acceleration and displacement as boldface to remind you that in this equation the signs of these vector quantities are still important. The symbol • may simply treated as meaning "multiply the quantities, including their signs". Later, when we take up 2 and 3 dimensional motion, that symbol will take on new meaning.

Equations 9, 11 and 12 are sometimes called "Galileo's equations of motion", but remember that Galileo didn't have the benefit of the algebraic notation we use here. The important point is that these equations follow from Galileo's definitions of velocity and acceleration by straighforward algebraic manipulation. Here, we see the power of mathematics to recast simple propositions into more useful forms: equations that we can use to analyze motion and to predict the outcomes of motion in particular cases.

Two-dimensional motion (in a plane).

Galileo extended kinematics to the cases of falling bodies and projectiles (cannonballs). He introduced an innovative concept to do this. He proposed that motion of a projectile could be considered as a combination of two simpler motions, combined by a "principle of superposition." It goes like this.

Suppose a cannon fires horizontally from a high cliff. The resulting motion includes forward motion and falling motion. Let's imagine that we can look at them separately. If the cannonball were simply dropped from the cliff, without any forward (horizontal) push and without any downward push, its initial velocity is zero, and equation 11 reduces to

[13]

y = at²/2

To keep things straight, we are considering the motion to be described in a Cartesian coordinate system with "x" being the horizontal distance, and "y" being vertical, with "+" being downward.

Now imagine "shutting off" the downward motion of the cannonball. Galileo realized that the amount of acceleration of a moving body depends on the angle of motion to the vertical, and that under these conditions a horizontally moving body does not accelerate or decelerate. He discussed this at length, for it's a tricky argument to make, considering that he had to set aside such complications as air drag, friction, etc. So Galileo said that such a motion, strictly horizontal, has acceleration of zero, and therefore equation 11 reduces to

[14]

x = v_ot

The body moves forward equal distances in equal times, at constant speed equal to the initial speed v_o.

Now, if both "falling" and "forward motion" are happening at once, Galileo said that we can combine (superpose) these two motions to determine the actual motion. Just decide on a value of time at which you'd like to know where the cannonball is, put that sinto equations 13 and 14, and you get the position of the body expressed in terms of the horizontal distance x from the starting point, and the vertical distance of fall, y, measured from the starting point. [We are expressing this in the language of Descartes' coordinate geometry, which was devloped later than Galileo, but we see the conceptual basis anticipated in the work of Galileo and others who were investigating the problem of motion. In fact, Galileo's analysis anticipated the important development of the vector concept in mathematics, not formulated fully and explicitly used in physics until the 19th century!

The diagram illustrates what's going on. It shows the horizontal and vertical displacements from the starting point at the upper left. Along the horizontal line, imagine arrows with their tails at the starting point. Their heads are shown. Arrows downward represent the displacements from this horizontal line, due to "falling". The lengths are drawn at equal time intervals, and in accordance with equations 13 and 14. The superposition of the horizontal and vertical motions is accomplished by adding the vertical and horizontal displacements for each time, combining the arrows by putting them head-to tail in a string of two. This string has one arrow "tail" at the starting point and the other arrow "head" at the position of the body at that time. The displacement of the body at that time is represented by an arrow (not shown) that extends from the starting point to the arrowhead we just constructed.

This procedure works correctly even for firing at other angles to the horizontal, as when a cannon fires at an upward angle. This procedure allows one to trace out the trajectory throughout the body's motion. Using the equations we have developed, this path will be the mathematical curve called a parabola. In the real world one must take into account the influence of air, retarding the motion, an effect which is more significant for large speeds. Unfortunatelly that complication would require an analysis using calculus.

Vector addition.

The process we have shown is a very important one in physics, so we might as well look at it with more care. These arrows we drew are representations of vectors, specifically displacement vectors. Many quantities in physics combine by a process of combining the vectors by vector addition the process of putting the "arrow" representation of the vector quantities in a head-to-tail string. The sum of those vectors is then the vector drawn from the free tail to the free head.

Scalars and Vectors

A scalar is a physical quantity whose definition does not in any way depend on direction in space. Scalars include time, mass, volume, temperature, density and others. The size of a scalar quantity is represented as a number. In physical equations, scalars obey the algebra of numbers.

Sum of three vectors.

A vector is a physical quantity dependent on direction in space. A vector has both size and direction, and both must be specified to uniquely characterize the vector. Vectors include displacement, velocity, acceleration, momentum, angular momentum and others.

Some vector quantities may surprise you. Consider a piece of surface area. Since surface area has orientation in space, it must often be treated as a vector, its direction being the line normal to the surface.

The direction of a vector quantity in physics must be specified, for it is as important as the vector's size in determining the vector's effect. Two vectors are said to be equal only if their sizes and directions are the same. When we write A = B we are saying that these two vectors are equal in size and in the same direction. The boldface equal sign tells us this is a vector equation. When we write A = B we are saying the two vectors are equal in size only, and the lightface equal sign represents a size equality only.

Vectors in physical equations obey an algebra quite different from that of scalar algebra. One useful and intuitive pictorial representation of a vector is an arrow. The length of the arrow represents the vector's size, and the direction of the arrow represents the vector's direction in space. With this representation we may illustrate relations between vectors geometrically, and this facilitates using geometry to solve some problems.

The size (or magnitude) of a vector quantity is a scalar, i.e., a number. The size of vector V is symbolized |V|. (Note that the V is boldface, since it is a vector, but the lightface "absolute value" brackets, | |, indicate that the entire quantity is a scalar. I.e., we take the absolute value of a vector, which gives a scalar result.) Alternatively we may simply write the vector's letter symbol lightface to indicate that it represents only the size of the vector.

The sum of two vectors may be found by geometrically placing the arrows representing the vectors head-to tail. The sum may then be found by drawing a new vector from the free tail to the free head.

The difference between two vectors A − B is found by adding the vectors A and −B where −B is a vector of the same size as B but opposite in direction.

The projection of a vector onto a line is Vcosθ, where θ is the angle between the vector and the line and V is the size of the vector. This may be found geometrically by constructing lines perpendicular to the reference line and from the head and tail of the vector. The length along the reference line lying between the construction lines is the "projection of the vector along that line".

Component of a vector on a line.

Components of vectors. When dealing with vectors algebraically it's useful to represent a vector by its components. The component of a vector is the projection of that vector onto a chosen coordinate axis. The component of a vector is usually treated as a scalar quantity. While the coordinate axis can be any line, we customarily use the axes of a Cartesian coordinate system, specifying the x, y, and z components of the vector. When this is done, any vector in the space is uniquely defined by specifying the coordinate axes, and the vector's components along those axes, (x,y,z). Sometimes it's even useful to have a set of coordinate axes that are not orthogonal (perpendicular), so long as the axes "span the space" (are able to uniquely represent any vector in the space by its components). This requirement is met if the three axes do not all lie in the same plane and no two are collinear.

Usually the vectors' tails aren't on a coordinate axis. Drop two perpendiculars to the axis, from the vector's head, and from its tail. Then the component on that axis is the length between the feet of these perpendiculars. The illustration shows vectors lying in an x,y plane, but the same principle is used with three dimensional situations.

Components on Cartesian axes.

The components of vectors are signed numbers; they may have positive or negative sign. Subtract the tail projection value from the head projection value to get that signed number.

If the components of two vectors are (x₁,y₁,z₁) and (x₂,y₂,z₂) then the sum of these three vectors can be written in component notation as (x₁+x₂,y₁+y₂,z₁+z₂).

The product of a vector and a scalar, Vs, is a vector of size |V|s. It has the same direction as V.

Two kinds of vector products are useful in physics, and these will be defined later when we need them.

Galileo's equations in vector notation.

As we have seen, vector quantities add in a a geometric way, quite different from the way ordinary numbers (scalars) add. Two vectors of size 3 and 5 do not usually addd to a vector of size 8. We get that result only when they happen to be in the same direction. But this process of vector addition may be represented by an extended algebra that includes vectors and scalars. We distinguish the vectors from scalars by using boldface for vector quantities and lightface for scalar quantities. When a mathematical operation symbol like "+" or "−" stands between two terms that are vectors, the symbols represent vector addition and subtraction of the sort we have just described. When such operators stand between scalars, they represent ordinary scalar addition of numbers. Two vector quantities are equal only when their sizes and directions are equal.

Let's see how this works out by trying to recast Galileo's kinematics equations in vector notation.

Definitions:

[21]

Velocity: v = (x₂ − x₁)/(t₂ − t₁), which may be written v = Δx/Δt

[22]

Acceleration: a = (v₂ − v₁)/(t₂ − t₁), which may be written a = Δv/Δt

The Δx and Δv quantities are now vector displacements and velocities respectively.

As before, we can write an expression for average velocity in the case where the acceleration vector is constant. Draw the picture representing equation 23 and 24.

The average of two velocities.

Average velocity: <v> = (v₁ + v2)/2

And <v> = d/t

These two may be immediately combined to give the equation

[25]

(v₁ + v₂)/2 = d/t

for any two points 1 and 2.

Initial conditions: The body has displacement d = 0 at t = 0, and its velocity at t = 0 is v_o. We sometimes refer to t = 0 and t as the "initial" and "final" velocities, but in fact they are simply the times at which the moving body happens to be at two different positions, chosen because they are of particular interest to us in problem-solving. The equations below are results valid only for the case of constant acceleration.

[26]

Velocity: v = d/t

[27]

Acceleration: a = (v - v_o)/t

Use equation 25 written for the "initial" and "final" velocities v_o and v. Then solve the equation for d.

d = (v_o + v)t/2

Solve equation 27 for v

v = v_o + at

Put this into equation 29.

[31]

d = v_ot + (at²/2)

When we were doing straight line motion we multiplied equation 7 by equation 9 to obtain the final kinematic equation. But now that would require multiplying a vector by a vector, something we haven't yet defined. So it's time to define the dot product of two vectors:

A • B = A B cos θ

where A and B are two vectors of size A and B, and θ is the angle between them when they are placed tail-to-tail. It follows that the dot product of a vector A with itself is its magnitude squared (A²). The dot product of two perpendicular vectors is zero, because the angle between them is 90°. We can now perform the dot product of equations 27 and 29 to get:

v² = v_o² + 2a•d

This may be written: v² = v_o² + 2 a d cos θ.

Vector quantities obey the distributive law of algebra, a fact used in deriving this last result.

Energy revealed.

Equation 33 is especially meaningful for the further development of physics concepts. In this and the next two sections we anticipate where this leads, by looking at how Newton's law builds upon the laws of kinemtatics. Suppose we multiply equation 33 by m/2, where m is the mass of the moving object.

(1/2)mv² = (1/2)mv_o² + ma•d

Newton's equation defines force by F = ma. This introduced force and mass into the picture, and the science of mechanics resulted by applying Newton's equation to kinematics. If we make this replacement in the last term we get:

(1/2)mv² = (1/2)mv_o² + F•d

Defining work as W = F•d, and rearranging the terms, we get:

W = (1/2)mv² − (1/2)mv_o²

This is the "work-kinetic energy theorem", W = ΔE_k, where E_k is the kinetic energy defined by E_k = (1/2)mv².

Momentum revealed.

But there's more to be revealed from Galileo's kinetmatic equations, when Newton's law is combined with them. Look at equation 27.

a = (v − v_o)/t

Multiply this by mt.

mat = mv − mv_o

Replace, using F = ma.

Ft = mv − mv_o

We have arrived at the law relating impulse to momentum, Ft = Δp where Ft is defined to be impulse and p = mv is defined to be the momentum.

Comments about energy and momentum.

We have demonstrated how the laws of momentum and energy and the definitions of work and impulse arise naturally from kinemtaic laws combined with Newton's dynamic law, F = ma. A more general treatment would consider how this plays out in the less restrictive cases where the acceleration isn't constant. This treatment would require calculus, but it would arrive at the same expressions for impulse, work, energy and momentum.

Students sometimes wonder whether both energy and momentum are necessary in mechanics. Don't they both describe motion, and use the same "raw materials"? In the early history of physics, this question occupied some great minds for a long time, before it was clear that both are absolutely necessary. Kinetic energy is a scalar quantity and depends on the square of the velocity, so it cannot by itself tell us about the directions of moving bodies. Momentum is a vector quantity depending on the first power of velocity. But it also doesn't tell us everything about the motion. It is true that sometimes a physics problem may be solved for the desired quantity by using only energy, or only momentum. But many problems require the simultaneous application of both equations. Even such a simple problem as a ball bouncing from the floor. See the document ball bouncing from a massive wall.

Galileo's law of inertia and his method of superposition.

The above presentation has been in the modern language of vectors. Let's return to Galileo's methods to see how he used his "method of superposition", and to what extent his work anticpated later developments. Textbooks seldom present this as Galileo did. A good web document about this is The Pendulum Swings again: A Mathematical Reasessment of Galileo's Experiment with Inclined Planes by Alexander Hahn.

Galileo's Principle of Inertia: A body moving on a level surface will continue in the same direction at constant speed unless disturbed.

We could nitpick this. What is "level" and how do you test whether the surface is level? A very large flat surface on the earth presents problems, for a body placed on it would undergo simple harmonic motion due to the fact that the gravitational force acts always toward the center of the earth (not everywhere perpendicular to a flat surface). Friction counts as a "disturbance" acting to slow the speed of the body. But such quibbling aside, Galileo certainly was anticipating Newton's first law here.

Galileo's Principle of Superposition: If a body is subjected to two separate influences, each producing a characteristic type of motion, it responds to each without modifying its response to the other.

Again we could quibble about precise language, but I think that Galileo was describing what we would now describe with vectors. One must of course, first establish that the motion being studied does obey a superposition principle. Superposition is not a universal principle. Many phenomena have separate influences that do not combine in this way. Today we would say that to obey this kind of superposition principle, two influences must combine linearly and must be independent. These are tricky qualifications to explain. "Linear" means that they mathematically superpose (add) by scalar or vector addition. The influences A and B combine by some sort of mathematical sum, A + B. They do not combine by A² + B², for example, or by some other complicated mathematical combination such as A + B + 2AB. "Independence" means that one influence does not modify the fundamental nature of the other. The presence of influence B does not change the equation for influence A.

Galileo used a principle of supsrposition in several important ways.

Superposition of displacements.

A nice demonstration illustrating the Galileo superposition method is the "monkey and hunter" experiment. A monkey dangles from a tree limb. The hunter aims his rifle directly at the monkey. The monkey has not had a physics course, so he thinks that he can avoid getting shot by letting go of the limb when he sees or hears the shot. But the hunter has had physics, which is why he aimed directly at the monkey, counting on the startled monkey to drop from the limb when he hears the sound of the gun. The hunter knows that the bullet will hit the monkey anyway, on the way down. Two web sites describe this: Oberlin College and San Francisco U. The explanation is often described poorly in textbooks. But with the principle of superposition it is easy. The bullet's displacement is the superposition of v_ot directly along the line from gun to the initial position of the monkey. The monkey, at nearly the same instant the bullet leaves the gun, falls according to (1/2)gt². The bullet falls the same amount in the same time, and therefore will hit the monkey. Draw the vectors v_ot and (1/2)gt² head-to tail to see this. You know this problem has been around many, many years, for nowadays we don't want to encourage shooting monkeys.

Superposition of two identical systems.

Galileo used a variant of superposition to show that bodies of different mass falling the same distance must cover that distance in equal times.

• Two completely identical bodies falling the same distance will fall in equal times.
• They will still fall in equal times if falling beside each other.
• Even if they touch as they fall, they will still fall in equal times.
• Glue or fasten them together where they touch, they will still fall in the same time, but now the composite body weighs twice as much.
• Therefore the size of the mass doesn't affect the fall.

Of course this assumes idealized conditions, where the surrounding air has negligible influence on the motions.

A similar argument shows that the pendulum's period is independent of the mass of the pendulum bob.

• Two completely identical pendulums have the same period.
• They will still have the same period if swinging beside each other.
• If they are brought close together, even touching, they will still have the same period.
• Glue or fasten them together where they touch, they will still have the same period, but now the composite body weighs twice as much.
• Therefore the size of the mass doesn't affect the pendulum's period.

This sort of "logical" presentation of an argument is very "Aristotelian" in character. Yet here Galileo is cleverly using Aristotelian methods to reach conclusions contrary to those of Aristotle.

Galileo's relativity principle

Superposition of simultaneously acting influences is the basis of Galileo's method for relating relative velocities of three or more bodies. Consider an airplane doing 200 mph speed straight East in still air. But then it encounteres a sidewind of 30 miles per hour blowing southward. You wish to find the plane's speed and heading in the wind, assuming that the plane does not deliberately change its course.

First, label things carefully with paired subscripts:

Velocity of plane with respect to still air: V_pg = 200 mph East.
Velocity of wind (air) with respect to the ground: V_ag = 30 mph South.
Velocity of plane with respect to the ground: V_pg

Galileo's superposition principle for velocities may be written:

V_AB + V_BC = V_AC

Note that the pairing of the subscripts shows a pattern, to aid memory. In our problem, we renotate the subscripts:

V_pa + V_ag = V_pg

We don't even have to rearrange the terms in this equation. So now draw this vector equation as a vector polygon, which is a right triangle. The vector V_pg is the hypotenuse, and has size given by the Pythagorean theorem (200² + 30²)^1/2 = 202 mph. If the pilot makes no course correction, the new direction of flight is at an angle south of east, of size given from tanθ = 30/200. So θ = 8.53°.

We chose a simple example for illustration. The vector polygon will not generally be a right triangle. And sometimes to achieve the correct pairing of subscripts to match equation 39 one must rearrange the vector terms, remembering that when a vector is "moved across the equal sign" you must change its sign.

—Donald E. Simanek, Feb, 2005

Go to Notes on chapter 3
Go forward to the mechanics chapter.
Return to the contents page of A Brief Course in Classical Mechanics.
Return to Donald Simanek's front page.