Kinematics is a mathematical branch of physics that uses equations to describe the motion of objects (specifically *trajectories*) without referring to forces.

These equations allow you to simply plug various numbers into one of the four basic **kinematic equations** to find unknowns in those equations without applying any knowledge of the physics behind that motion, or having any knowledge of physics at all. Being good at algebra is sufficient to bludgeon your way through simple projectile-motion problems without gaining a real appreciation for the underlying science.

Kinematics is commonly applied to solve **classical mechanics** problems for motion in **one dimension** (along a straight line) or in **two dimensions** (with both vertical and horizontal components, as in **projectile motion**).

In reality, events described as occurring in one dimension or two dimensions unfold in ordinary three-dimensional space, but for kinematics purposes, x has “right” (positive) and “left” (negative) directions, and y has “up” (positive”) and “down” (negative) directions. The concept of “depth” – that is, a direction straight toward and away from you – is not accounted for in this scheme, and it usually does not need to be for reasons explained later.

## Physics Definitions Used in Kinematics

Kinematics problems deal with position, velocity, acceleration and time in some combination. Velocity is the rate of change of position with respect to time, and acceleration is the rate of change of velocity with respect to time; how each is derived is a problem you may encounter in calculus. In any case, the two fundamental concepts in kinematics are therefore position and time.

More on these individual variables:

- Position and displacement are represented by an
**x, y coordinate system**, or sometimes *θ* (Greek letter theta, used in angles in geometry of motion) and *r* in a polar coordinate system. In SI (international system) units, distance is in meters (m). - Velocity
*v* is in meters per second (m/s). - Acceleration
*a* or

*α*

(the Greek letter alpha), the change in velocity over time, is in m/s/s or m/s^{2}. * Time t is in seconds. * When present, initial and final

**subscripts** (

*i* and

*f*, or alternatively,

*0* and

*f* where

*0* is called "naught") denote initial and final values of any of the above. These are constants within any problem, and a direction (e.g.,

*x*) may be in the subscript to provide specific information as well.

Displacement, velocity and acceleration are **vector quantities**. This means they have both a magnitude (a number) and a direction, which in the case of acceleration may not be the direction in which the particle is moving. In kinematic problems, these vector in turn can be broken down into individual x- and y-component vectors. Units such as speed and distance, on the other hand, are **scalar quantities** as they have a magnitude only.

## The Four Kinematic Equations

The math needed to solve kinematics problems is not itself daunting. Learning to assign the right variables to the right pieces of information given in the problem, however, can be a challenge at first. It helps to determine the variable the problem asks you to find, and then look to see what you are given for this task.

The four kinematics formulas follow. While "x" is used for demonstrative purposes, the equations are equally valid for the "y" direction. Assume constant acceleration *a* in any problem (in vertical motion this is often *g*, the acceleration owing to gravity near Earth's surface and equal to 9.8 m/s^{2}).

Note that (1/2)**(v** **+** **v _{0})** is the

*average velocity*.

This is a restatement of the idea that acceleration is difference in velocity over time, or a = (v − v_{0})/t.

A form of this equation where initial position (y_{0}) and initial velocity (v_{0y}) are both zero is the free-fall equation: **y = −(1/2)gt**^{2}. The negative sign indicates that gravity accelerates objects downward, or along the negative y-axis in a standard coordinate reference frame.

This equation is useful when you don't know (and don't need to know) time.

A different kinematics equations list might have slightly different formulas, but they all describe the same phenomena. The more you lay your eyeballs on them, the more familiar they will become even while you are still relatively new to solving kinematics problems.

## More About Kinematic Models

Kinematic curves are common graphs showing position vs. time (*x* vs. *t*), velocity vs. time (*v* vs. *t*) and acceleration vs. time (*a* vs. *t*). In each case, time is the independent variable and lies on the horizontal axis. This makes position, velocity and acceleration *dependent variables*, and as such they are on the vertical axis. (In math and physics, when one variable is said to be "plotted against" another, the first is the dependent variable and the second the independent variable.)

These graphs can be used for **kinematic analysis** of motion (to see in which time interval an object was stopped, or was accelerating, for example).

These graphs are also related in that, for any given time interval, if the position vs. time graph is known, the other two can be quickly created by analyzing its slope: velocity vs. time is the slope of position vs. time (since velocity is the rate of change of position, or in calculus terms, its derivative), and acceleration vs. time is the slope of velocity versus time (acceleration being the rate of change of velocity).

## A Note on Air Resistance

In introductory mechanics classes, students are usually instructed to ignore the effects of air resistance in kinematics problems. In reality, these effects can be considerable and can slow a particle greatly, especially at higher speeds, since the *drag force* of fluids (including the atmosphere) is proportional not merely to the velocity, but to the square of the velocity.

Because of this, any time you solve a problem including velocity or displacement components and are asked to omit the effects of air resistance from your calculation, recognize that the real values would likely be somewhat lower, and time values somewhat higher, because things take longer to get from place to place through air than the basic equations predict.

## Examples of One- and Two-Dimensional Kinematics Problems

The first thing to do when confronting a kinematics problem is identify the variables and write them down. You can, for example, make a list of all of of the known variables such as x_{0} = 0, v_{0x} = 5 m/s and so on. This helps pave the way for choosing which of the kinematic equations will best allow you to proceed toward a solution.

One-dimensional problems (linear kinematics) usually deal with motion of falling objects, although they can involve things confined to motion in a horizontal line, such as a car or train on a straight road or track.

**One-dimensional kinematics examples:**

*1. What is the final velocity of a penny dropped from the top of a skyscraper 300 m (984 feet) tall?*

Here, motion occurs in the vertical direction only. The initial velocity **v**_{0y} = 0 since the penny is dropped, not thrown. y – y_{0}, or total distance, is -300 m. The value you seek is that of v_{y} (or v_{fy}). The value of acceleration is –g, or –9.8 m/s^{2}.

You therefore use the equation:

This reduces to:

This works out to a brisk, and in fact deadly, (76.7 m/s)(mile/1609.3 m)(3600 s/hr) = 172.5 miles per hour. IMPORTANT: The squaring of the velocity term in this type of problem obscures the fact that its value may be negative, as in this case; the particle's velocity vector points downward along the y-axis. Mathematically, both **v** = 76.7 m/s and **v** = –76.7 m/s are solutions.

*2. What is the displacement of a car traveling with a constant velocity of 50 m/s (about 112 miles per hour) around a race track for 30 minutes, completing exactly 30 laps in the process?*

This is a trick question of sorts. The distance traveled is just the product of speed and time: (50 m/s)(1800 s) = 90,000 m or 90 km (about 56 miles). But displacement is zero because the car winds up in the same place it starts.

**Two-dimensional kinematics examples:**

*3. A baseball player throws a ball horizontally with a speed of 100 miles an hour ( 45 m/s) off the roof of the building in the first problem. Calculate how far it travels horizontally before hitting the ground.*

First you need to determine how long the ball is in the air. Note that despite the ball having a horizontal velocity component, this is still a free-fall problem.

First, use **v** **= v _{0} + at** and plug in the values v = –76.7 m/s, v

_{0}= 0 and a = –9.8 m/s

^{2}to solve for t, which is 7.8 seconds. Then substitute this value into the constant velocity equation (because there is no acceleration in the x direction)

**x = x** to solve for x, the total horizontal displacement:

_{0}+ vtor 0.22 miles.

The ball would therefore in theory land close to a quarter of a mile away from the base of the skyscraper.

## Kinematics Analysis: Speed vs. Event Distance in Track and Field

In addition to supplying useful physical data about individual events, data pertaining to kinematics can be used to establish relationships between different parameters in the same object. If the object happens to be a human athlete, there are possibilities for using physics data to help chart out athletic training and determine ideal track event placement in some cases.

For example, the sprints include distances up to 800 meters (just shy of a half-mile), the middle-distance races encompass the 800 meters through about the 3,000 meters and the true long-distance events are 5,000 meters (3.107 miles) and above. If you examine the world records across running events, you see a distinct and predictable inverse relationship between race distance (a position parameter, say *x*) and world-record speed (*v*, or the scalar component of *v*).

If a group of athletes runs a series of races across a range of distances, and a speed vs. distance graph is created for each runner, those who are better at longer distances will show a flatter curve, as their speed slows less with increasing distance compared to runners whose natural "sweet spot" is in shorter distances.

## Newton's Laws

Isaac Newton (1642-1726) was, by any measure, among the most remarkable intellectual specimens humankind has ever witnessed. In addition to being credited as being a co-founder of the mathematical discipline of calculus, his application of math to physical science paved the way for a groundbreaking jump in, and lasting ideas about, translational motion (the kind under discussion here) as well as rotational motion and circular motion.

In establishing a whole new branch of classical mechanics, Newton clarified three fundamental laws about the motion of a particle. **Newton's first law** states that an object moving at constant velocity (including zero) will remain in that state unless perturbed by an unbalanced outside force. On Earth, gravity is virtually always present. **Newton's second law** asserts that a net external force applied to an object with mass compels that object to accelerate: **F _{net}** = m

**a**.

**Newton's third law** proposes that for every force, there exists a force equal in magnitude and opposite in direction.

References

- BC Open Textbooks: Time, Velocity, and Speed
- Georgia State University: HyperPhysics: Description of Motion
- Physics LibreTexts: Problem-Solving for Basic Kinematics
- International Association of Athletics Federations: World Records
- Vanderbilt University: Escape Velocity
- Georgia State University: HyperPhysics: Newton's Laws

About the Author

Kevin Beck holds a bachelor's degree in physics with minors in math and chemistry from the University of Vermont. Formerly with ScienceBlogs.com and the editor of "Run Strong," he has written for Runner's World, Men's Fitness, Competitor, and a variety of other publications. More about Kevin and links to his professional work can be found at www.kemibe.com.