### Concept Overview

- Kinematic variables: displacement, distance, velocity, speed, acceleration, and time.
- Kinematic Equations
- Average speed vs average velocity
- Motion in one dimension – cars, balls, and any object moving in one direction
- Motion in two dimensions – object dropped from a moving plane, a bullet fired horizontally, balled kicked at 45°, and all types of projectile motion.
- Graphs: displacement vs time, velocity vs time, acceleration vs time.

### Kinematic Variables

There are 5 total kinematic variables: ∆x, v_{0}, v_{f}, a, t.

Velocity (v) is split into two variables: Final Velocity (v_{f}) and Initial Velocity (v_{0}).

Time is measured in second and doesn’t have a direction.

Now let’s cover some misconception.

#### Displacement (∆x)

**Displacement** (a vector quantity) is how far an object has moved from its origin. ** Distance** (a scalar quantity) is the total amount traveled.

Suppose you took a 200-mile trip to LA and back. Your displacement is zero (because you’re back where you started). Your distance, however, is 400 miles (200 miles to and 200 miles back)

#### Velocity (v_{0} and v_{f})

**Velocity** (a vector quantity) is how fast displacement changes. **Speed** is how fast distance changes.

Suppose you run 1 lap around a 100 meter track in 100 seconds. Your speed is 1 m/s. And your velocity is 0 m/s.

#### Acceleration (a)

**Acceleration** (a vector quantity) is how fast velocity changes. Think of you speeding up or slowing down in your car. An acceleration can also be caused by a change in direction. Kinematics ONLY deals with uniform (constant) acceleration. Non-uniform acceleration requires calculus.

Suppose you drop a ball from a tall building. How fast is it traveling after 3 seconds?

Since the ball accelerates due to gravity at 9.81 m/s^{2}, this means that every second the ball gains 9.81 m/s of speed. So after 3 seconds it reaches a speed of 29.4 m/s.

**Time** (a scalar quantity) is often overlooked. It’s the key to solving many projectile motion (2 dimensional) problems. Why?

Because time is the SAME in the X and Y direction (more on why this matters below).

### Kinematic Equations

Couple of things to remember regarding variable notation:

**∆**= delta. It means “change in,” which literally means “final – initial.”**s**= distance.**∆x**= change in horizontal displacement.**∆y**= change in vertical displacement**v**= initial velocity. It is pronounced “V knot.”_{o}- Equations apply both horizontally and vertically
- Use subscripts in the equations to keep track of direction. For example, initial velocity in the vertical direction can be written as, “v
_{oy}“. Displacement in the vertical direction can be written as ∆y.

#### Memorize

Memorizing the big 4 kinematic equations goes a long way. It only takes around 10 minutes.

Students that have memorized the formulas, can solve kinematic problems 30-50% faster than students that haven’t.

The best way to memorize is to write them down on a flashcard and practice 20 kinematic problems.

Here are the big 4 kinematic equations (last one is an extra):

- v_f = v_0 +at
- v_f^2 = v_0^2 + 2a\Delta x
- \Delta x = v_0t + \frac{1}{2}at^2
- \Delta x = v_ft - \frac{1}{2}at^2
- \Delta x = \frac{1}{2} t (v_f + v_0)

### Average Speed vs Average Velocity

While we already covered velocity above, this topic seems to confuse students a bit. So here’s a more in depth explanation.

There is big difference between average speed and velocity.

* Average speed* is distance divided by time: speed = \frac{d}{t} .

** Average velocity** is displacement divided by time: \overline{v} = \frac{\Delta x}{\Delta t} .

Example: suppose you run 100 meters north, then back 100 m south, in 100 seconds. Using the equations from above we can find that

a) your average speed is = 200 m ÷ 100s = 2 m/s

b) your average velocity is 0m ÷ 100s = 0m/sec. (Remember that displacement is 0, since you end back up at your starting position)

### Motion in 1-Dimension

Any object moving in a straight line ( up, down, left, right ) is moving in one direction. A car that starts from rest and accelerates to 80 m/s in 30 seconds, is moving in one direction.

Let’s say we are asked to find the distance the car traveled in 30 seconds. Start by choosing a kinematic equation that has all the given variables, then plug the numbers in. It’s as simple as that!

#### Picking the Right Equation

Here’s a little trick if you can’t figure out what equation to use:

There are no more than four variables in each kinematic equation. Similarly, every kinematics * problem* has four variables: three known and one unknown variable)

Write out the 4 variables and see which kinematic equation has all 4. In the example above we have: v_{o} , v_{f} , t , and ∆x. Thus you should use the equation: \Delta x = \frac{1}{2} t (v_f + v_0) .

### Motion in 2-Dimensions

Objects moving in 2-dimensions, are moving **both horizontally and vertically**. These objects are called projectiles. Solve problems the exact same way as with one-dimensional kinematics, except apply the formulas in both directions.

#### Helpful Tips to Remember

- Time stays the same in both the x and y directions
- Horizontal speed, distance, or time, can be found using ∆x = vt
- The vertical acceleration is almost always, acceleration due to gravity (9.81 m/s)
- Unless given, the initial velocity is almost always 0.
- Vertical velocity at the top of the trajectory is always 0, but horizontal velocity remains constant.
- If a projectile is traveling at an initial angle, use trig to split the vector into horizontal and vertical components.
- The final velocity of the object is the SUM of the horizontal and vertical components of velocity

#### Solving 2d Motion Problems (Example)

A bullet shot horizontally at 90 m/s, from a height of 3.2 meters. Find how far the bullet travels.

To solve this problem just figure out what’s given in the horizontal and vertical directions, and pick a kinematic equation to solve for the unknown. Here’s an example of what the work might look like:

Horizontal Components | Vertical Components |
---|---|

v_{ox} = 90 m/s | v_{oy} = 0 m/s |

t =? | ay = 9.81 m/s^{2} |

∆x =? | ∆y = 3.2 m |

t =? |

Hopefully, you can see the pattern here. We aren’t given enough information in the horizontal direction. Therefore, we will use a kinematic equation in the vertical direction to find the time.

Remember that **time** is the common factor in both directions. If we solve for t, we will be able to solve for the distance the bullet travels with the equation \Delta x = v_{0x} t .

### Motion Graphs

Most students overlook graphing. It comes up quite a bit on the AP Exam. I’ll show you the best way to understand how these graphs work. First lets try a problem:

From the acceleration-time graph below, find the total change in velocity in the first 3 seconds.

First, let’s take a step back and go back to definitions

- Velocity is the change in displacement per unit of time. “Change in” is another word for slope/derivative! Thus the slope of the displacement vs time graph is velocity!
- Acceleration is the change in velocity per unit of time. Thus the slope of the velocity vs time graph is acceleration.

Look at it like this: **∆x → ∆v → a.** To get to the next one just find the **slope**/derivative!

Now go backwards: **a → ∆v → ∆x.** To get to the next one just find the opposite of the derivative: the integral (a fancy way of saying **area** under the curve!)

Are you able to solve the problem above now? If you are, great job! Keep practicing with the problems at the end of this post!

### Practice Questions for Mastery

^{2}. What was the car's initial velocity?

589 m

755 m

964 m

1005 m

1127 m

.92 seconds

1.8 seconds

2.4 seconds

3.3 seconds

3.9 seconds

acceleration and velocity are both zero.

the ball has zero velocity for a split second.

the ball’s range is at a maximum.

both the x and y components of velocity are zero.

the x component of velocity is greater than the y component of velocity.

53 m/s

41 m/s

1681 m/s

776 m/s

more information is needed.

## Quick Answers

- 0 m/s
- 18.6 m/s
- (d)
- (b)
- (e)
- (a)
- It takes the same time for both.
- 200 m/s

### Bonus – Challenge Questions

These are much harder questions, typically seen on an AP Exam.

constant

increasing

decreasing

dependent on the height

unknown

*h*when fired straight up. If the same gun is pointed at an angle of 45° from the vertical, what is the new maximum height of the projectile?

*h*/4*h*/(2√2)*h*/2*h*/√2*h*

## Quick Answers

- (c)
- (b)

### Free Practice Tests

Here are some free kinematics practice test made by other sites! These are AP level questions.

- Test #1
- Test #2
- Test #3 (Beginner Level)
- Test #4 (AP Level)
- Test #5 (AP Level)
- Free Response Questions

### Conclusion

Congrats! That wraps up everything you need to know for kinematics in both regular and advanced Physics classes.

It’s best to ** apply** your knowledge to see if you really are understanding these concepts.

So if you are still having trouble answer these questions, try practicing more questions. Or try working with a teacher or professional physics mentor to iron out any misunderstanding.

Up next, we have a speed review for linear forces!