Graphical Analysis of Motion

Graphical Analysis of Motion

     Distance vs Time or Displacement vs Time or Position vs Time

• The slope always represent the speed on a distance vs time graph.
• High slope = faster speed
• Low slope = slower speed
• A straight line  means constant speed
• slope = rise/ run = 5/2 = 2.5 m/s = constant speed
• slope = 5 m/ 2 s = 2.5 m/s

Example: 

     If a rock is dropped from a high cliff. The rock fall for 5 sec, how far did it traveled?

     

d = si*t + (1/2)a t^2     si = 0 (initial speed)    t = 5s    a = 9.8 m/s^2

d1 = 1/2 (9.8) (1s)^2 = 4.9 m

d2 = 1/2 (9.8) (2s)^2 = 19.6 m

d3 = 1/2 (9.8) (3s)^2 = 44.1 m

d4 = 1/2 (9.8) (4s)^2 = 78.4 m

d5 = 1/2 (9.8) (5s)^2 = 122.5 m

    

   

• A constant acceleration will create a curve line (parabola).
• The slope at any given point is the speed or velocity at that instant
• The instantaneous speed is found by drawing a tangent line to any point on the curve line.
• slope = Δd / Δt 

 Speed vs Time graph

• Area below the line on a speed vs time graph is the distance traveled.
• Area = L * W => d = s * t =  3 s * 10 m/s = 30 m

• Straight line means constant acceleration = slope = (6 m/s) / 5 s = 1.2 m/s^2
• Initial speed =0, Distance traveled =>d = (1/2)at^2 = (1/2) (1.2)(5)^2 = 15 m
• Area = 1/2bh = (1/2)(5)(6) = 15 m 

Velocity vs Time graph

• Slope is always = accelerate on a velocity vs time graph
• In this case the slope is negative, so it is decelerate.
• Slope = (0-140)/ (8-0) = - 17.5 m/s^2
• Distance = Area of the line => A = (1/2)bh = (1/2)(8)(140) = 560 m
• Distance = d = si*t + (1/2) at^2 = (140)(8) + (1/2) (-17.5) (8)^2 = 560 m

Acceleration vs Time graph

• Straight line means constant acceleration = 10 m/s^2 , but velocity is changing by the same rate each second
• Area under the line gives the change in velocity during the time interval 5 secs. 

Area = (10 m/s^2 )(5s) = 50 m/s =  Δv (change in velocity)

Formula:

1) d = si*t = (1/2) a*t^2     a = (2d) / t^2

2) d = (sf^2 - si^2) / 2a     a = (sf^2 - si^2) / 2d

3) a = (sf - si) / t     t = (sf -si) / a

    _

4) s = d / t     t = d / s     d = s*t

    _

5) s = (si +sf) /2

6) sf = si + a*t

Kinematics: Study of motion

Includes the concepts of:

• speed
• velocity
• acceleration
• time
• distance
• displacement

Vectors: is a quantity which express both magnitude and direction.

• magnitude is how big or how small a number is
• direction is which way something is headed or directed.
• example; velocity, displacement, force, accerleration

Scalars: is a quantity which express magnitude only 

• no direction is specified
• example; speed, mass, distance, energy

Distance: how far something has traveled

• no direction is specified
• commonly measure in meters, kilometers, and centimeters
• is a scalar

Displacement: how far something is from its starting point

• direction is specified
• is a vector

 

Example: An object travel 20 meters west then 30 meters north.

                  a) What is the distance traveled?

                  b) What is the displacement?

Use Pythagorean Theorem: a^2 +b^2 = c^2

                    

                    a) Distance = 20 m +30 m = 50 m

                    b) Displacement = 20^2 +30^2 = x^2

                                                     √ (1300)   = x^2

                                               36.06 m = x (North West)

Speed: how fast something is traveling

• speed is the distance covered in a given amount of time
• is a scalar

   The formula for speed: s = d/t, s= speed, d = distance, t = time

    

Example 1) An object covered a distance of 75 kilometers in half an hour. What is the speed in a) kilometers/hour b) meters/second

     a) s = d/t = 75 kilometers / .5 h = 150 km/h

    

    b) (150 km)/ h * (1000 m)/ km * h/(3600 s) = 41.67 m/s 

Example 2) An object covered 450 m in 20 s. What is it speed in a) m/s b) km/h?

     a) s= d/t = 450 m/20 s = 22.5 m/s

     

     b) (22.5 m)/ s * km/ (1000 m) (3600 s)/h) = 81 km/h

Example 3) A rifle fire a bullet at a target. The speed of the bullet is 700 m/s. The target is located 250 meters away.

     a) How long does it take for the bullet to hit the target?

    b) What is the bullet speed in km/h?

     a) s = d/t => 700 m/s  = (250 m)/ t => 250 m*(1s/700 m) = (t/250m) *250 m => 250/700 = t, t = .35 secs.

   b) (700 m)/ s * (3600 s)/h * km/ (1000 m) = 2520 km/h

Velocity: how fast something is traveling and in what direction

• is a vector

Example:

     a)  An object travel 45 m/s is a speed

     b)  An object travel 45 ms due south is a velocity

Acceleration: how fast speed or velocity changes

• the rate at which speed or velocity changes
• if an object speed up, it has a positive acceleration
• if an object slowing down, it has a negative acceleration

The formula to calculate the acceleration is: a =  Δ's / t

• Δ's is the change in speed, Δ is Delta
• t is the time
• the change in speed is calculate using sf - si
• sf = the final speed
• si = the initial speed
• therefore a= (sf - si)/ t

Examples 1) A new Nissan GT-R 35, Spec V acceleration from 0 mph to 60 mph in 3.2 secs. What is it rate of acceleration?

     a = (sf - si) /t = (60 - 0) / 3.2 = 18.75 (m/h)/s, mph = miles per hour. So in 1 sec, the car go 18.75 miles per hour.

Example 2) A car goes from 20 m/s to 50 m/s in 10 s. What is the rate of acceleration?

     a = (sf - si) /t = (50-20)/ 10 = 3 m/s^2 

         (m/s) /s => m/s x 1/s = m/s^2

The average speed of a uniformly acceleration object:

Example:  If a car goes from 10 m/s to 30 m/s in 3 s.

     a) What is its acceleration rate?

        

         a = (sf - si)/t = (30 - 10) / 3 = 6.67 m/s^2

     b) What is it average speed over the 3 s interval?

                                                                        -

         The formula to calculate average speed: s = (si + sf) /2

= (10 + 30) / 2 = 40/2 = 20 m/s

Rearranging acceleration formula: a = (sf - si) /t 

• First multiply both side by t => a*t = sf - si
• Final speed of an object base on initial speed, acceleration and time. => sf = si +a*t

Example 1) An airplane traveling at 300 m/s accelerates uniformly at the rate of 12 m/s^2. What its speed be at the end of 30 secs?

     si = 3000 m/s     a = 120 m/s^2     t = 30 s     sf = ?

   sf = si + at = 300 m/s + (120 m/s^2) (30 s) 

                     = 300 m/s + 360 m/s = 660 m/s

Example 2) An automobile accelerate at the rate of 7.5 m/s^2. It goes from 15 m/s to 85 m/s. While it is accelerating, how long did it take to reach 85 m/s ?

     

     a = 7.5 m/s^2     si = 15 m/s     sf = 85 m/s     t = ?

     sf = si + at => 85 m/s = 15 m/s + (7.5 m/s^2) t

                             70 = 7.5 t, t =9.33 s

Distance based on si, a, and t: To calculate how far 
an object travels 

based on initial speed, acceleration rate, and time.

The following equation is used => d = si*t + (1/2) a t^2

• d = distance traveled
• si = initial speed
• a = acceleration rate
• t = time

Example 1) A rocket is accelerate from 300 m/s at 
the rate of 9.5 m/s^2 for 20 sec. 

     

     a)What is the rocket final speed?

     

          si = 300 m/s     a = 9.5 m/s^2     t = 20s

          

          sf = si + at = 300 m/s + 9.5 m/s^2 (20s)

                            = 300 m/s + 190 m/s = 490 m/s

     b) What is the distance traveled by the rocket during the 20 sec?

          d = si*t + (1/2) a t^2 = 300 m/s (20s) = (1/2)(9.5 m/s^2)(20s)^2

                                           = 6000 m + 1900 m  = 7900 m

Example 2) A rock is dropped off a bridge. Gravity 
accelerates falling objects at the rate of  9.8 m/s^2.

     a) If the rock fall for 5 sec, how far did it traveled?

          si = 0     a = 9.8 m/s^2     t= 5s

          d = (0) (5s) + (1/2)(9.8 m/s^2) (5s)^2

         

             =    0        + (9.8 x 25)/2 = 122.5 m

     b) How fast was it going at the end of 5 secs?   

          sf = si + at = 0 + (9.8 m/s^2) 5s =  49 m/s 

The last kinematics formula: 

     a = (sf^2 - si^2) / 2d          d = (sf^2 - si^2) / 2a

• sf = final speed
• si = initial speed
• d = distance traveled
• a = acceleration rate

Example 1) A supersonic jet travelling at 400 m/s accelerate to 750 m/s at a rate of 18 m/s^2. What distance does it cover during this time?

     si = 400 m/s     sf = 750 m/s      a = 18 m/s^2

     d = [(750 m/s) ^2 - (400 m/s)^2] / 2 (18 m/s^2) = 11, 180.55 m

Example 2) A race car travels 800 m while acceleration at the rate of 5 m/s^2. If its initials speed was 30 m/s. What was it final speed?

     sf = si + at     a = (sf^2 - si^2)/ 2d

     5 m/s^2 = [sf^2 - (30 m/s)^2 ] / 2 (800 m)

     1600 (5) = [(sf^2 - 900)/ 1600] (1600)

      8000 + 900  = sf^2 -900 +900

   

           8900       = sf^2

          

           √8900     = sf^2

     

          94.34 m/s = sf 

Trigonometric Functions

Trigonometric Functions

a) right triangles , triangles that contain a 90° angle.

b) adjacent side always makes up of θ (Theta) and the hypotenuse side

Example:

            1)

                 Tangent 42° = O / A = x / 25 m 

     

                           25 (.9004) = (x / 25) (25)

                              22.51 m =  x

             2)

                                

                              Sine 35° = O / H = x / 18 m

                    18(.5736) = (x / 18)(18)

                   10.3248 m = x

                     

Scientific Notation

Scientific Notation

A way of expressing very large  or very small number using power of 10.

     coefficient = a, base = 10, exponent = b

Write a number in scientific notation:

Example: 

1) 125,000,000,000

     a) Put a decimal after the first digit and drop the zeros; 1.25 will be the coefficient

     b) To find the exponent, count the places from the decimal to the end of the number; 11 is the exponent

     c) So, the scientific notation is 1.25 x 10 ^11

2) 0.000000000000244  = 2.44 x 10^-13 

(Numbers smaller than 1 will have a negative exponent)

3) 360,000,000,000,000  = 3.6 x 10^14

4) 0.000000297              = 2.97 x 10^-7

5) 125,000,000               = 1.25 x 10^8

6) 1,380,000                   = 1.38 x 10^6

7) .000,006                     = 6.0 x 10^-6

8) .0000274                    = 2.74 x 10^-5

Adding and Subtract using Scientific Notation:

Examples:

     1)   3.6 x 10^5             2)       8.4 x 10^8

          +                                     -

           4.5 x 10^5                       3.2 x 10^8

           -------------                      -------------

           8.1 x 10^5                       5.2 x 10^8

     3)   4.5 x 10^6        =>         0.45 x 10^7

          +                          

           2.2 x 10^7                      2.2   x 10^7

                                     ^           --------------

                                     ||           2.65 x 10^7

                                     ||

          a) increase the smaller exponent to equal the larger exponent

          b) decrease the size of corresponding coefficient amount

     4)   3.3 x 10^9                      3.3   x 10^9

          +                       =>       +

           2.2 x 10^8                      0.22 x 10^9

                                                  --------------

                                                  3.52 x 10^9

     5)   8.4 x 10^5                      8.40  x 10^5

          -                       =>        -

           3.4 x 10^4                      0.34  x 10^5

                                                  -------------

                                                  8.06  x 10^5

    6)    3.3 x 10^-7                     3.30 x 10^-7

          +                     =>           + 

           8.6 x 10^-8                     0.86 x 10^-7

                                                  ---------------

                                                  4.16 x 10^-7

    7)    3.6 x 10^-9                     3.60 x 10^-9

          -                     =>           -

           4.4 x 10^-10                   0.44 x 10^-9

                                                 ----------------

                                                  3.16 x 10^-9

Multiply and Divide using Scientific Notation: 

 A) When multiply, multiple the coefficients with the powers of ten, you add the exponent.

(a x 10^b) * (c x 10^d) = (a*c) x 10^(b +d)

Example 1: 

(5.2 x 10^2) * (2.5 x 10^3) = (5.2*2.5) x 10^(2+3) = 13 x 10^5  (Proper scientific notation is 1.3 x 10^6)

B) When divide, divide the coefficient with the powers of ten, you subtract the exponent.

(a x 10^b) / (c x 10^d) = (a/c) x 10^(b-d)

Example 1: 

(6.3 x 10^4) / (2.1 x 10^2) = 6.3/2.1 x 10^(4-2) = 3 x 10^2

Example 2: 

(8.49 x 10^-6) / (3.0 x 10^-3) = (8.49/3.0) x 10^(-6-( -3)) = 2.83 x 10^ -3 (Negative exponent!)

Example 3: 

(8 x 10^3) (2 x 10^2)        

-------------------------   =   [(8)(2)] /4  x 10^[ (3+2)-3]  =  4 x 10^2

      (4 x 10^3)                      

                                                        

Scientific Method

(Then...)

Scientific Method

A systematic procedure for solving problem in science.

The steps in the scientific method:

1. Problem Statement
2. Research
3. Hypothesis
4. Procedure
5. Data and Observations
6. Conclusion

There are also theories and laws.

1. Problem Statement: What are you trying to accomplish in the experiment.

2. Research: Find out about the topic, gather information and references related to the problem.

3. Hypothesis: An educated guess as to what you think will happen in an experiment.

    

     Example: I believe that the dog will be flattened. (When a heavy weight dropped on it).

3. Procedure: The steps that you will follow in executing the experiment.

4. Data and Observation: What you observe in an experiment. Includes all pertinent characteristic and measurement.

5. Conclusion: A statement of what happened and whether or not this agrees with your hypothesis.

    Example: My hypothesis was correct. The dog got flattened.

6. Theory: An explanation for what happened in the experiment.

7. Law: A scientific fact that has withstand the test of time and repeated experimentation.

     Example: Law of gravity - All object are attracted toward the center of the Earth.

Metric System

Metric System Prefixes

Metric System

Based on multiples of 10. Prefixed are place in front of the base unit to represent larger or smaller quantities.

Important prefixes:

     kilo (K)      = 1,000
     hecto (H)   = 100
     deca (D)   = 10
     Base         = 1 (meter, gram, liter, joule)
     deci (d)     = .1 (1/10)
     centi (c)    = .01 (1/100)
     milli (m)    = .001 (1/1,000)

Other prefixes:
     mega (M)  = 1000,000
     giga (G)     = 100,0000,000
     micro (µ)   = 1/1000,000

Base Units:

• meter (m) is the base unit for length
• gram (g)     "        mass
• liter (l)        "       volume
• joule (J)      "       energy

Examples:

   1) Convert 30 mL to kL

         a) 30.0, move the decimal to the left six places,  = .000030 kL

          OR 

         b)

    2) How many grams are there in 5 kilogram (kg) ?

         a) 5.0, move the decimal to the right three places, = 5000. g

OR

         b)

    3) Convert 350 Dg to mg.

          a) 350.0, move the decimal to the right four places, = 3500000. mg

          OR

          b)

      4) Convert 20 cm to m.

          a) 20.0, move the decimal to the left two places, = .20 m

          OR

          b) 

Measurement of matter

     Mass: the amount of matter within an object
     
     Weight: the force of gravity of mater acting on an object (9.8 m/s^2)

     Volume: the amount of space taken up by an object.
                  
                   Volume = A x h, A = area, h = height

High School Physics

Physics: is the scientific study of matter and energy and how they interact with each other.

• Matter is anything that has mass and volume (takes up space)
• Energy is the capacity or ability to do work; work - the apply force over a given distance

State of Matter: 

• Solid:  
• rigid and have a definite shape and volume.
• particles vibrate more slowly than in a liquid   
• particles are very close together and in fixed position, not compressible
• Liquid:
• flow, take the shape of its container 
• definite volume, no definite shape
• particles vibrate faster and are even farther apart than in a solid
• particles slide past each other freely, not compressible
• Gas:
• expand to fill container
• no definite shape or volume
• particles are widely spaced apart
• not compressible
• Plasma:
• made up of superheated particles
• affected by electric and magnetic fields
• Superfluid ( such as Bose-Einstein Condensate):
• made up of  particles cooled to temperature nears absolute zero (0 Kelvin)
• opposite of plasma
• not naturally occur