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Linear

Motion

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Kinematics

Kinematics is the mathematical treatment of the motions of bodies without regard to the forces that produce the motion.

Measuring Length

Distance and displacement are two quantities which may seem to mean the same thing, yet they have distinctly different meanings and definitions.

Distance ( d ) – the total length of a path that an object travels without regard to direction. (scalar)

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Displacement ([pic]) – the change in the position of an object from its starting point to finishing

point. (vector)

To test your understanding of this distinction, consider the motion depicted in the diagram below. A physics teacher walks 4.0 meters East, 2.0 meters South, 4.0 meters West, and finally 2.0 meters North.

| | |

|Distance – 12.0 meters |Displacement - 0 meter |

Determine the distance and displacement if the physics teacher walked 4.0 meters East and then 2.0 meters South.

Determine the distance and displacement if the physics teacher walked 4.0 meters East.

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| | |Round Trip |

| |Running from Home to 7-Eleven | |

| |1.2 miles |3.0 miles |

|Distance | | |

| |~ .8 miles E |0 mile |

|Displacement | | |

Describing Motion in Words

The motion of objects can be described by words - words such as distance, displacement, speed, velocity, and acceleration. These mathematical quantities which are used to describe the motion of objects can be divided into two categories. The quantity is either a vector or a scalar. These two categories can be distinguished from one another by their distinct definitions:

• Scalar – a quantity having magnitude (amount) only, with no direction specified.

Ex – Time (30 seconds), Mass (45 kilograms)

• Vector – a quantity having both magnitude and direction.

Ex - Velocity (25 m/s, due north)

Relative Motion

Motion –a change in the position of a body with respect to time.

• Frame of Reference – the perspective (point of view) of an observer.

o Ex – You are a passenger in a car traveling on the Long Island Expressway. You are observing the other cars moving around you. You are in Car #1 traveling at 25 m/s as shown in the picture below.

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Questions

1. For Cars #2, #3, and #4, state the direction the car will appear to be moving with respect to your car. Also determine the relative speed of the car.

| | | |

|Car #2 |Car #3 |Car #4 |

| | | |

|5 m/s ↑ |0 m/s |10 m/s ↓ |

2. You are now magically transported into Car #4. For Cars #1, #2, and #3, state the velocity and direction YOUR car will appear to be moving with respect to the other cars.

| | | |

|Car #1 |Car #2 |Car #3 |

| | | |

|10 m/s ↓ |15 m/s ↓ |10 m/s ↓ |

Based upon the above exercise and video, construct a summary statement about determining the motion of objects.

Motion is relative, meaning it is dependent on the motion of the observer. Therefore in order to know the true motion of an object the observer must know their own motion.

Ex – Ray’s Zuvlight is traveling in his ’78 Corvette at a speed of 50 m/s. Approaching him from the opposite direction is Willie Makit who is driving his ’71 Mustang also at a speed of 50 m/s.

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50 m/s 50 m/s

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Questions

1. If you were a passenger in Ray’s car, what would be the relative speed of Willie’s car?

100 m/s

2. If you were an innocent bystander how fast would each car appear to be moving?

50 m/s

Speed and Velocity

• Speed ( v ) – the distance that an object moves in a unit of time. (scalar)

• Velocity ( v ) – the displacement of an object in a given amount of time. (vector)

Linear motion refers to an object’s change of position within two dimensions. On a straight path, there are only two possible directions for velocity.

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Varying Velocities

Usually applied to an object not traveling at constant velocity

Instantaneous Velocity (speed) – the velocity (speed) of an object at any particular instant in time

Average Velocity – the rate at which an object approaches its destination.

Now let's try a more difficult case by considering the motion of that physics teacher again. The physics teacher walks 4.0 meters East, 2.0 meters South, 4.0 meters West, and finally 2.0 meters North. The entire motion lasted for 24 seconds. Determine the average speed and the average velocity.

Average speed = _____0.50 m/s____

Average velocity = _____0 m/s____

Solving Problems

GIVENS UNKNOWNS EQUATION SUBSTITUTE SOLUTION

1. Marion Jones is able to run 200. meters in 23.4 seconds. What is her average speed?

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2. What is the average speed of Nicholas Cheep if he travels 6.00 meters North in 2.00 seconds and then travels 3.00 meters East in 1.00 second?

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3. A bus travels 280 km south along a straight path with an average velocity of 70. km/h to the south. How long does the total trip last?

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Acceleration

Acceleration

• Acceleration ( a ) - a vector quantity representing a change in an object’s velocity over a period

of time.

Note –this course only deals with uniform acceleration.

Velocity and Acceleration

Directions - For each example below, describe the car’s motion.

1. A car is driving down a long straight section of the LIE on cruise control at 55 mph.

Constant Speed? Constant Velocity? Constant Acceleration?

2. A car speeds up smoothly on a straight entrance ramp to the LIE in preparation for merging onto the highway.

Constant Speed? Constant Velocity? Constant Acceleration?

3. A car slows down smoothly in preparation for driving off an exit of the LIE.

Constant Speed? Constant Velocity? Constant Acceleration?

4. A driver pumps the brakes while trying to slow down during a rainstorm.

Constant Speed? Constant Velocity? Constant Acceleration?

5. A driver maintains a speed of 35 mph while exiting on a curving exit ramp.

Constant Speed? Constant Velocity? Constant Acceleration?

Calculating Average Speed of an Accelerating Object

The average speed of an object accelerating uniformly can be calculated by using the equation:

This formula is valid only when the acceleration is constant.

• Example:

o A particle is accelerated uniformly from a speed of 10. m/s to 50. m/s in 5.0 seconds. Find the average speed of the particle during this 5.0 – second interval.

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Types of Acceleration

You do a good deed by picking up a blind hitchhiker and driving them home. The hitchhiker notices along the way that you accelerated the car in three different ways. What are they and how did the hitchhiker know it? What are the three instruments in your car that control each of these types of acceleration?

1) speeding up 2) slowing down 3) changing direction

- gas / accelerator - brakes - steering wheel

Calculating Acceleration

1. A car accelerates from rest to a speed of 36 m/s in 4.0 seconds.

a) What is the car’s average acceleration?

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b) What is the car’s average speed?

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2. A runner moving at 2.0 m/s increases his speed to 3.0 m/s as he approaches the finish line. The time for this increase in speed was 2.1 seconds. What is the sprinter’s acceleration?

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3. A car accelerates from rest at a rate of 5.5 m/s2. How much time will it take to reach a speed of 34 m/s?

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4. A motor boat took 3.2 seconds to slow uniformly from 6.0 m/s to 4.0 m/s as it approached the return lane when coming back into shore. What is the boat’s acceleration?

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Negative Acceleration

A tractor drives east at 15 m/s into a hurricane which is blowing west. The strong wind slows the tractor down at a rate of 5 m/s/s. Fill in the data table below showing the velocity and acceleration of the tractor for the first 6 seconds of travel.

|Time(s) |0 |1 |2 |3 |4 |5 |6 |

|Velocity (m/s) |15 m/s |10 m/s |5 m/s |0 m/s |-5 m/s |-10 m/s |-15 m/s |

|Acceleration (m/s2) |- 5 m/s2 |- 5 m/s2 |- 5 m/s2 |- 5 m/s2 |- 5 m/s2 |- 5 m/s2 |- 5 m/s2 |

Compare the directions of the velocity and acceleration for each part of the trip. Now, use this

comparison to make a conclusion regarding whether the tractor is speeding up or slowing down.

When velocity and acceleration have opposite signs (direction), the object slows down. When they have the same sign (direction), the object speeds up.

• Can an object have acceleration but have a velocity of zero? Give an example.

Yes, in the case of the tractor, though it was constantly changing velocity, at one moment it was at zero.

• Can an object have a velocity but have an acceleration of zero? Give an example.

Yes, when an object is traveling at constant speed.

• Is negative acceleration the same thing as deceleration? Explain.

Sometimes. Deceleration is the same as negative acceleration when acceleration and velocity are in opposite directions.

Kinematics Equations

Solve the following questions using proper significant figures and the GUESS method.

1. A motorcycle traveling at 12.6 m/s accelerates at a rate of 1.7 m/s2 for 3.4 seconds. What is its final velocity?

vf = vi + at = ( 12.6 m/s) + (1.7 m/s2)(3.4 s) = 18.4 m/s

2. A bullet is accelerated from rest at a rate of 400 m/s2 for 0.05 seconds. How far did it travel while it was accelerating?

d = vit + ½ at2 = (0 m/s) + ½ (400 m/s2)(.05 s)2 = .5 m

3. An elephant accelerates from 5 m/s to 10 m/s at a rate of 2.0 m/s2. What is the elephant's final displacement?

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4. A driver brings a car traveling at 22 m/s to a full stop in 2.0 seconds.

a) What is the car's acceleration?

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b) How far did the car travel before stopping?

d = vit + ½ at2 = (22 m/s)(2.0 s) + ½ (-11 m/s2)(2.0 s)2 = 22 m

5. A jet stops in 525 m using a constant acceleration of -8.0 m/s2. How fast was it moving initially?

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6. Skid marks left from a stopped car are 27 meters long. If the car had a deceleration of 6.0 m/s2 and stopped in 3.0 seconds, how fast was the car moving initially?

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7. Starting from rest, a lion moves 110 m in 5.0 seconds. What was the acceleration of the lion?

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8. Theoretically, a person wearing a seatbelt can withstand a deceleration of -300. m/s2. A test dummy is placed in a sled and experiences this deceleration over a distance of 1.5 m. How fast was the sled moving initially?

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For a character to move in a flip book, its position must be different from one page to the next.

DVD – Frame of Reference #1

DVD – Frame of Reference #2

Have students create their own problems.

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Speed Pop-Up Video

t = 1.5 s

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Harry Potter Pop-Up Video

t = 9.5 s

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People run this fast

Reference Table uses symbols interchangeably

Positive

Up

Right

East

North

forwards

Negative

Down

Left

West

South

backwards

+

-

+

-

Not just the average of speeds!!!

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Givens

d = 200. m

t = 23.4 s

Unknowns

v = ?

Givens

d[pic][?]$%?ž¯°A B M N üíáÕƳ᫣œ?zn_SKC?4h®^ãtot = 9.00 m

ttot = 3.00 s

Unknowns

v = ?

f = final

i = initial

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Givens

v = 36 m/s

t = 4.0 s

Unknowns

a = ?

Givens

vi = 0 m/s

vf = 36 m/s

Unknowns

vavg = ?

Givens

vi = 2.0 m/s

vf = 3.0 m/s

t = 2.1 s

Unknowns

a = ?

Givens

v = 34 m/s

a = 5.5 m/s2

Unknowns

t = ?

Givens

vi = 6.0 m/s

vf = 4.0 m/s

t = 3.2 s

Unknowns

a = ?

vf = vi + at

vf2 = vi2 + 2ad

d = vit + ½ at2

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