Kinematics is the study of the motion of objects. In this chapter, we will introduce you to the concepts of motion, and to the mathematical ideas needed to perform motion equations.

Distance is an important part of physics. We often need to make measurements or calculations based on how far away something is. Distance is a simple concept- it is a measurement of the space between two things.

But physicists also need to refer to an object’s position. Position requires what is known as a frame of reference. Instead of describing a point in terms of it’s distance from another point, we can describe it’s location in terms of it’s distance AND direction from a reference point. In this example, "Zero" is the reference point, and the line that the points are on is the frame of reference.

It is important to know that is we moved the "Zero" point, then the position of Point A would change. Yet the distance between the points does not change.

Position is what we call a Vector quantity, and Distance is a Scalar quantity. Scalar quantities have only a value, or a magnitude. An example of a scalar measurement could be “2 meters”. Vector quantities, however, need a magnitude, a direction and a reference point. An example of a vector measurement could be “2 meters to the right of zero”. Sometimes a value can have a positive (+) or a negative (-) value. These values are also Vectors, because they have magnitude and direction. You should learn and remember the difference between vectors and scalars.

When an object is in motion, it is said to have a Velocity. In order to determine how fast an object is moving, we need to know two things.

What is the object's Displacement?
How long has the object been in motion?

Displacement is very similar to Distance. When an object is at rest (not moving) it has a definite position. Then, after it has moved, it has another definite position. The distance between the starting and ending positions of the object is called the Displacement. Because Displacement can be positive (+) or negative (-) it is a Vector quantity.

Displacement is usually given the symbol

d.

Time is easy- it is simply the amount of time (in seconds) that the object was moving. Since time has no direction, it is a Scalar value.

Time is usually given the sybmol

t.

If you think about it, velocity is just a measurement of how far an object move in an amount of time. If it moves a long way in a short time, it is going very fast. If it moves a short distance in a long, long time, then it has a low velocity.

Therefore, we can say that the formula for Velocity is:

v =

d /

t

The unit of velocity is the meter per second (writen as m/s) but sometimes Kilometers per Hour are used.

Remember, our formula can be re-written using basic algebra. It can be used to solve for displacement or for time.

d = v

t =

d / v

Click on the Java logo and you will be taken to the home page of a Physicist who creates Java applets. You will be able to run his simulation to get a better idea of the relationship between distance, velocity and time.

Example Problem

A train travels at +227 km/h for 2 hours. There are 280 people on board. All of them are named . How far did it travel?

Given:
v = + 227 km/h

t = 2 h

Equation:

d = v

t

Answer:

d = ( +227km/h )( 2h ) = 454 km

Velocity, however, does not stay the same all the time. During the course of an objects motion, the velocity can be increasing or decreasing. For example, a plane could be going 30m/s in the first 10 seconds of its motion on the runway, then after the 20 seconds, it could be going 60m/s, then after 30 seconds, the plane is going 90m/s. For every ten seconds, the velocity increases 30m/s. Therefore, the plane is going 3m/s/s – that is, the plane increases its velocity by 3m/s every second. The plane is accelerating.

Acceleration is the ratio between the change in velocity and the change in time. The change in velocity is simply the difference between the final velocity and the initial velocity. The change in time is just how long the acceleration continued for. Therefore, average acceleration is equal to:

vf = Final Velocity
vi = Initial Velocity
a = acceleration
t = time

a = vf - vi / t
or
a =

v /

The plane above is experiencing constant acceleration. Acceleration that does not change in time is what is meant by constant. The formula that describes the relationship between the initial velocity, the final velocity, the acceleration, and the time of all this motion is:

vf = vi + at

Example Problem

If a car with a velocity of 2.0 m/s accelerates at a rate of 4.0m/s/s, what is its velocity after 2.5 seconds?

Equation:

vf = vi + at

Answer:

vf = 2m/s + (4m/s/s)(2.5s) = 12m/s

vf = 12m/s

When an object is moving with constant acceleration, the distance it traveled can be found by multiplying its velocity by how long it has been traveling. Therefore, we can manipulate the following equation to find out distance, velocity, or time:

d = ( vf + vi)t / 2

When you know what the acceleration and time are, you can calculate displacement another way. Now, here is where the math you learned at the beginning comes in. We have to manipulate the equations we just learned into one equation that will find displacement when we know time and acceleration. We know vf = vi + at and d = ( vf + vi)t /2 so we can do the following:

d = (vf + vi)t /2

Substitute vi + at for vf
d = ((vi + at) + vi)t /2

Simplify equation
d = (2vi + at)t /2

Multiply inside of bracket by t

Our new formula:

d = vit + (0.5)at^2

In some instances, we know velocity and acceleration. Then to find distance, another formula is needed. Actually, like we did above, we will have to substitute equations, and make up a new one:

d = ( vf + vi)t /2

USE ALGEBRA TO FIND t

t = vf - vi/a
d = (0.5)( vf + vi)( vf - vi)/a
d = (0.5)(vf2 - vi2) /a

Our new formula:

vf^2 = vi^2 + 2ad

Enough formulas for now, don’t you think! Dont worry if you had trouble understanding how we got all of these equations. Try reading the lesson again! Since there are so many equations, there is a list of them at the end of this lesson. Our Online Reference section also has all of these equations, but the key is to only remember one or two then learn how to manipulate them. Here are some practice questions to help you remember how to use these:

Example Problem

An airplane must reach a velocity of 71m/s for takeoff. If the runway is 1 km long, what must the constant acceleration be? **Hint change km’s to m’s.**

Given:

You know the initial velocity which is 0, the final velocity which is 71m/s, and the distance which is 1 km or 1000m (use meters because that is the standard).

You know 4 formulas, and three of them use time. Since you have no idea how long it takes to takeoff, you need to use the last one.

Answer:

Substitute the numbers in:

(71m/s)2 = (0m/s)2 + 2(a)(1000m)

Open the calculator (press the button underneath the logo) and solve. The acceleration is equal to 2.5 m/s/s.

Graphs are important in physics because of the information we can get from them. A graph is a physicists way of drawing a picture. It is the means of showing a relationship. And it is an excuse to doodle in class. But, most importantly, like formulas, we can manipulate graphs to tell us the information we need to solve a problem.

Distance VS Time

With a distance vs time graph, we can find out how much displacement has occurred at a given time very easily. But, we can also find the velocity or if there had been any acceleration at the time.

Velocity: You must find the slope in order to find velocity. The slope of the graph in a distance vs time will give you the velocity. Why? Because the slope is RISE over RUN. The distance is plotted according to the Y-axis (vertical) and the time is plotted along the x-axis (horizontal). Rise is distance, run is time. Rise is meters, run is seconds. Rise over run is meters per second. So, the slope is the constant velocity.

Acceleration: If there is a curve in the distance-time graph, that means that the velocity is not constant. It means that there is an acceleration happening. This is best solved by drawing a Velocity vs. Time graph, which we will now tell you about.

Velocity VS Time

When you have a velocity-time graph, there is more that you can do. The slope is the acceleration, and a curve means that the acceleration is changing. But, you can find distance as well. Take a look at the graph below:

To find the distance, just calculate the area under the graph. To find the area you multiply the length (x-axis) by the height (y-axis) (you also divide by 2 for a triangle) and the you get the distance. You get distance because you are multiple m/s by s which gives you m.

Example Problem

You can use graphing to solve problems that stumped people for thousands of years. No, really, try this one: (You may have to read it a bunch of times in order to understand it.)

Given: Phred was a really fast runner, and Phil was a slow tub of lard. However, Phil was a really clever person. One day he bet that if Phred gave him a head-start of only 5 meters, Phil would win the race. Phred quickly accepted the bet, knowing that he could beat Phil, even if Phil had a 20m head start.

Fat Phil laughed and said that he wouldn’t even bother to run, because Phred would never be able to win. Phil’s logic was that if Phred gave him a head-start, he would never be able to catch up. He explained that after Phred ran the five meters to catch up to Phil, Phil would be ahead to a new point- lets say at 9m. Then Phred would keep running four meters to catch up, but within that time Phil would be even further ahead. Phred would continue to run, but by the time he made it to the position Phil was in, Phil would have just moved a little ahead.

Perplexed by this logic, Phred decided not to run, and he lost the bet. We all know Phred could have beaten Fat Phil. Why is this logic wrong?

This paradox literally stumped people for thousands of years. The solution is simple if you graph two speeds on a position time graph. At one point, the second runner will overtake the first, because they can both occupy the same position at the same time.

If you take a bowling ball and a tennis ball and drop them, which one will hit your foot first? Phil Physicist thinks the bowling ball will hit his foot first since it is heavier. Phrank Physicist hopes the tennis ball will since it is lighter and needs less force to accelerate it. Who is right?

Actually , they are both wrong. All objects fall to earth with a constant acceleration, if we ignore air resistance. (The air gets in the way of the falling objects.)

Acceleration due to gravity is given a special symbol – g. Since acceleration is a vector quantity it can either be positive or negative. Since acceleration due to gravity causes objects to have a velocity going down, the velocity will be negative. Therefore, the acceleration due to gravity is also negative.

g is always given the value -9.8m/s/s.

Any free falling object has an acceleration of 9.8m/s/s (and is negative to show it is accelerating downwards).

The formulas we use with acceleration due to gravity are the same with normal acceleration, but the ‘a’ in the formulas, is now ‘g’. So, just to recap, the formulas for acceleration or acceleration due to gravity are:

vf = vi + gt

d = vit + ½gt2

vf2 = vi2 + 2gd

Example Problem

A brick falls freely from a high building, landing on the head of some innocent person named . What is its velocity after 4.0s? How far does the brick fall in this time?

Given:

You know what the initial velocity is – 0m/s. The a brick is falling, so ‘g’ is -9.8m/s/s. The time is 4.0s. lug the numbers into your equation:

Work:

vf = 0 + (-9.8m/s/s)(4.0s)

= -39.2m/s (The negative means that the velocity is downwards)

Then, to solve for the distance, you know the final velocity, so you have the choice as to which formula to use. Let’s use the d = vit + ½at2. Just plug in the numbers!

Answer:

d = (0)(4.0) + ½(-9.8m/s/s)(4.0s)(4.0s)

= -78 m