Friday, December 30, 2016

Friday Gen Eds MS11: The Physics of Motion

This is the eleventh post in the math/science part of my "Gen Eds in a Nutshell" series. The Gen Ed series consists of ten subjects you might study in a general education or "liberal arts" core at a university or college. I've already done the subject of philosophy, and I'm over half way through the world history subject on Wednesdays. I'm combining the last two on math and science into one series on Fridays.

Thus far in the math/science subjects:
1. Physics is the study of the "laws" of nature as they relate to matter and energy. We can roughly divide it into 1) mechanics, which has to do with motion and its related forces, 2) thermodynamics, which has to do with heat and its associated dynamics, 3) electromagnetism, and what I might call 4) fundamental physics, such as relativity and quantum mechanics.

Mechanics also roughly divides into two basic types of discussion: 1) the description of motion (kinematics) and 2) the forces of motion (dynamics). This entry has to do with the first--kinematics, the description of motion. The next entry will deal with the forces of motion.

Straight Line Motion
2. If you have been driving for a while, you will know that if you are driving 60 miles/hour for an hour, you will have driven 60 miles. We can put this into a formula: d = vt. You can find the distance you have traveled by multiplying the velocity at which you are traveling by the amount of time you are going at that velocity.

There are some details we might mention. First, you have to use the right units for this equation to work. So if you are using "miles per hour" for the velocity, you have to use hours for the time. Otherwise, you're not ready to multiply the two together. What you're trying to do is cancel out the "per hour" part with the number of hours. So if you had minutes or seconds in one place and hours in the others, it wouldn't cancel right.

A physicist might also want to clarify that there is technically a difference between speed and velocity. When we talk about "speed," we don't care what direction you're moving in. Who knows, you might be driving back and forth between two points for an hour. In physics, when we talk of velocity, we are talking about speed in a specific direction. [1]

We might also make the formula a little more detailed by adding a starting point d1, the place where you started. Then the formula is d2 = d1 + vt .

3. Acceleration is the change in velocity over time. We know it from driving in a car or taking off in a plane. So if speed is distance per time, then acceleration is distance per time per time or distance per time squared. Just as we easily derived an equation for distance in relation to velocity, we can imagine a similar one for velocity in relation to acceleration"

v2 = v1 + at

So if you had an acceleration in meters/second squared and you multiplied it by seconds. One of the seconds in the bottom would cancel with the seconds you were multiplying by. And you would be left with meters/second, which is a velocity.

4. If you understood calculus and integration in the previous entry, we can integrate this equation to get another equation for distance traveled when an object is not moving at a constant speed but at a constant acceleration. It turns out to be:

d2 = d1 + v1t + 1/2at2

Another equation finds the distance by multiplying the average velocity by the amount of time. It looks like this:
Finally, if we take this last equation and move the variables around to come up with an equation for t and then substitute this into the second equation above, we end up with the last of the primary motion equations of physics:

5. These are the primary equations for motion in a straight line, such as if you were driving on a straight road, a line in what we might think of as a horizontal direction. A falling body is another kind of "straight-line motion." As it turns out, when a body is relatively close to the surface of the earth, the acceleration due to gravity is constant.

The acceleration of any body near the surface of the earth is 9.8 meters/second squared (or 32 feet/second squared). If we give this constant (meaning a value that doesn't change) the symbol g , we can express the equations above in relation to a body falling in a straight line.

So the distance something will drop in a given amount of time (treating the initial distance and velocity as zero) is

d = 1/2gt2

Implied in this equation is the fact that all objects fall at the same rate of acceleration, no matter how much they weigh. [2]

6. Projectile motion is a motion in two dimensions. If you hit a baseball into the air, there is motion going on in two different directions, driven by two different forces. First, there is a motion in a horizontal direction, an "x" direction. Then there is a motion in the vertical or "y" direction. Gravity is the force at work in the y direction. The force of the bat drove the ball initially both forward and upward.

You can analyze the initial trajectory of the ball in terms of these two components, the x and y components, using trigonometry. So you hit the ball at a certain angle from the horizontal. If you know the velocity at which the ball leaves the bat at this angle, then you can find the "component" of that velocity that is vertical and the component of that velocity that is horizontal.

The "sine" of that angle is the ratio of the opposite side or y component to the hypotenuse or overall velocity. The "cosine" of that angle is the ratio of the adjacent side or x component to the hypotenuse. The bottom line is that if you multiply the sine of the angle by the overall velocity, you will know the initial vertical velocity component for the ball. And if you multiple the cosine of that angle times the initial velocity, you will know the initial horizontal velocity component for the ball.

Now you can analyze these two components of the ball's trajectory separately. For the horizontal component of the ball's trajectory, we have Newton's first law, which we will discuss in the next post: "a body in motion wants to stay in motion." So the ball would continue forward indefinitely if it could (leaving wind resistance out of consideration for the moment).

What makes the ball's time in the air limited is the fact that gravity will pull the ball back down to the ground. By using the velocity equations above, we can figure out first how much time it will take for the ball's upward motion to stop and for it to start falling back down (v2 = v1 - gt), remembering that v2 will be zero. We know v1the initial upward velocity. We know g which is -9.8 m/s (negative because it's pulling downwards). So we can solve for t, the time in the air.

We can then find the distance the ball will travel upward by using the equation d2 = d1 + v1t - 1/2gt2. We now know t. We know the initial vertical velocity. We know g. d1 is however far above the ground the bat hit the ball. After we solve for d2, then we know how high the ball will go before it starts to fall back down.

Circular Motion
A special kind of motion is circular motion. In popular language, we talk about "centrifugal force," which we relate to the way a body wants to fly off of a merry-go-round or the way our bodies crash into the car door if the driver takes two sharp of a curve or the way a rock flies off a sling if you are spinning it around.

Technically, this is not a force but Newton's first law in action--a body in motion wants to stay in motion. Our body wants to continue in the same direction off the merry-go-round or into the car door or off the sling. Therefore, in order to stay on our circular trajectory, we need a force pulling us toward the center of the circle, called a "centripetal" force.

It turns out that a constant acceleration toward the center of a circle is necessary to keep an object moving in constant circular motion. If you spin a yo-yo in a circle, you must constantly apply a centripetal force with your hand. The formula for that constant "radial" acceleration equals v2/r, where r is the radius of the circle and v is the velocity of the object as it would fly tangentially off its circular path if you were to let go.

Periodic Motion
Another kind of motion that has become extremely important for our understanding of electromagnetism and the quantum world is periodic motion. This is a recurring motion such as when a weight is bouncing back and forth on a spring. It is fundamental to clocks of many kinds, such as when a pendulum swings back and forth or the old watches with springs in them. This back and forth motion is called "oscillation."

Some basic terms here are the "frequency" of the back and forth motion (e.g., how many "cycles per second") and the "period" of one back and forth motion (how much time it takes for one cycle). There is also the "amplitude" of the cycle, how big the displacement is from the resting point.

As it turns out, periodic motion can be quantified in terms of circular motion, since the back and forth happens in a cycle, like a point traveling around a circle. The relationship in its simplest form turns out to be:

x = A cos ωt

In this equation, x is the amount of displacement from the rest point at any time t. A is the amplitude or the maximum displacement. "cos" means the cosine of ωt. ω is then something called the "angular frequency." It is 2π times the frequency (cycles per second). [3]

Next Week: Math/Science 12: The Forces of Motion

[1] We say that velocity is a "vector" quantity because it implies motion in a particular direction, while speed is a directionless "scalar" quantity.

[2] Of course not taking into account the fact that something like a feather will experience an air resistance that will make it fall slower than a truck. However, in a vacuum, a feather and a truck would fall at the same acceleration.

[3] The reason for the 2π is to get the frequency into "radian" units, which measure a circle in terms of how many radiuses along the circle you are. The angular frequency ω is thus measured in radians traveled per second.

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