Linear Motion
Guiding Question
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How can we depict the motion of an object traveling in a straight line?
What factors contribute to the motion of an object in free fall?
Motion is Relative
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Question: If you are sitting still in a driving car, are you moving?
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Answer: It depends on your frame of reference. Relative to the car you are not in motion, you have a fixed position within the car . However, relative to your surroundings (the road and other objects you are driving past) the car and thus you are both in motion. These surroundings are not inherently at rest either. The Earth, to which they are attached is moving relative to the sun faster than you could ever drive (about 30 kilometers per second)!
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Think: If you are driving down the highway at 70 mph and a car passes you going 75 mph moving in the same direction. Will the car appear to move fast or slow relative to you and your car?
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Speed
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Speed Defined: Speed is a measure of how fast an object changes position, or the rate at which a distance is covered. The term rate indicates 'per unit of time', so speed is the distance covered per unit of time. The word per means "divided by", thus the formula for speed is distance/time.
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Example: km/h is read as "kilometers per hour" and is a measure of speed, the number of kilometers you travle in one hour.
Two Kinds of Speed
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Instantaneous Speed
how fast something is moving at any given moment.
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Measured by: Speedometer
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Example: If a car is going 50 km/h it may sustain that speed for one minuet or it may sustain it for an hour. If the car sustained that speed for one hour it would travel 50 km, but if it only sustained it for one minuet it would travel less than 1 km
Average Speed
how fast something gets from one point to another.
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total distance covered
average speed = --------------------------------
time interval
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Example: If you are driving to work you will not be traveling the same speed the entire commute, there are curves, stoplights, and different speed limits. The amount of time you set aside for the commute is not based on any given reading on your speedometer, but how long it will take you to get from where you are to where you're going.
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Velocity
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Speed vs. Velocity: speed and velocity are often used interchangeably in every day language, THIS IS NOT CORRECT! These two are similar but distinct.Velocity is the speed in a given direction.
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Think: Is it possible to have a constant speed but a changing velocity? Is it possible to have a changing velocity, but a changing speed?
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Lab: Can one person have the greater velocity while the other has the greater speed?
Now that we can define both speed and velocity we are going to run a short lab to determine whether having a greater speed necessarily means you have a greater velocity.
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LEARNING TARGETS
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speed is defined by time and distance
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velocity is defined by time, distance, and direction
Constant Velocity
must have both constant speed and constant direction​
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Example: moving in a straight line at a constant speed
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Changing Velocity
If the speed or direction changes, or both.​
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Example: the ball that has been falling down the left side of this page it moving in a straight line, but it is moving faster and faster as it falls so it's velocity is changing. Also, slowing down, making a turn, or moving in a circle will change your velocity
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Online Interactive Lab: The Moving Man Simulation – Constant Velocity Motion
You will use this online simulation to help you study the motion of an object with constant velocity.
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LEARNING TARGETS​
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Recognize graphs of position vs. time, velocity vs. time, and acceleration vs. time for motion with constant acceleration
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Determine the distance traveled from the area under the velocity vs. time graph
Acceleration
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Acceleration Defined: Acceleration is the rate at which velocity is changing, or how the velocity changes with respect to time
change in velocity final velocity - initial velocity
acceleration = -------------------------------- = -----------------------------------------
time interval time interval
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Units: In terms of base units, velocity is the rate of change in distance, for example meters per second. Now acceleration is the rate of change in velocity, or how quickly velocity is changing, so acceleration is meters per second per second. This can be written as meters per second squared (m/s^2)
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Example: a car siting still has no acceleration because it's velocity is not changing (it has no velocity), but a car traveling at a constant speed in a straight line has no acceleration for the same reason. If the same care were traveling at a constant speed along a curved path however it would have acceleration because the change in direction means a change in velocity.
The ball at the left side of the page has been accelerating down at the rate of free fall (more on that next). Thought it falls in a straight line with each time interval it moves a greater distance than in the previous interval, a change in velocity, thus it is accelerating
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Free Fall: How Fast
Online Interactive Lab: The Moving Man Simulation – Constant Acceleration Motion
You will use this online simulation to help you study the motion of an object with constant acceleration.
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LEARNING TARGETS​
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Recognize graphs of position vs. time, velocity vs. time, and acceleration vs. time for motion with constant acceleration
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Determine the acceleration from the slope of the velocity vs. time graph
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Determine the distance traveled from the area under the velocity vs. time graph
Question: If you drop a ball does it accelerate downward, or travel at a constant velocity?
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Answer: Well, we know the ball starts at rest and ends up in motion. Th ball has increased it's velocity and thus has accelerated. Also you would probably feel fine about catching a ball you dropped from one hand to the other, but less so about catching the same ball dropped to you from a tall building by your friend.
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Defining Free Fall: Gravity causes our ball to accelerate down when dropped. In real life air resistance affects the acceleration of a falling object, but if we remove air resistance then gravity is the only force acting on our falling object. When a falling object is affected only by gravity it is in free fall.
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Real Life Falling
gravity
air resistance
Free Falling
gravity
Question: How fast do things fall in free fall?
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Answer: With no air resistance, falling objects will be accelerated by gravity with no resisting force. On Earth acceleration due to gravity is 9.81 meters per second squared.
Note: during each second of the fall the instantaneous speed increases by an additional 9.81 meters per second
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change in velocity 9.81 m/s
acceleration = -------------------------------- = ---------------- = 9.81 m/s^2
time interval 1 s
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Free Fall: How Far
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Question: At the end of the first second a free falling object has an instantaneous speed of 9.81 m/s. Does this mean it falls 9.81 meters in that first second?
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Answer: No. Remember the difference between instantaneous speed and average speed? If an object's average speed over the course of one second is 9.81 m/s then it will travel 9.81 meters in that one second. but we know our object started from rest and took a full second to reach 9.81 m/s, so our average speed is between zero and 9.81 m/s.
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Question: What is the relationship between how fast something moves and how far it moves?
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Answer With Calculus: The derivative of a distance function is the instantaneous velocity, and the derivative of the velocity function is the instantaneous acceleration at a particular time
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Answer Without Calculus: Though these two concepts are related they are distinct from each other. to determine the distance an object fell in a particular time interval we calculate the average speed for that time. If we use the first second with our starting speed of zero and our final speed of 9.81 m/s...
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beginning speed + final speed 0 m/s + 9.81 m/s
average speed = -------------------------------------------- = ---------------------------- = 4.905 m/s
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so in the first second the object falls 4.905 meters. If we calculate the distance traveled in each subsequent second we will get this chart.
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Look at the free falling ball to your left. If the ball falls one unit of distance in one second then it will fall four units of distance in two seconds, and it will fall nine units of distance in three seconds. this is because the distance traveled is related to the square of the time.
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Think: How many units of distance will the ball fall in eleven seconds?
Lab: Galileo's Inclined Plane
Without high tech equipment, the motion of a falling object is difficult to measure. It all happens a bit too quickly for us to see what's going on. So what if we could slow down that motion? Galileo hit upon using incline plains to study the motion of falling objects. So instead of trying to watch objects falling he roll balls down these very gentle inclines so they built up speed very slowly.
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LEARNING TARGETS
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acceleration is the change in velocity through time
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study change in distance through time in a system of constant acceleration
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study change in velocity through time in a system of constant acceleration
Graphs of Motion
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Question: How do we graph the motion of an object?
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Answer: Speed vs. Time Graphs, Distance vs. Time Graphs, and Acceleration vs. Time Graphs
What happens in each of these graphs when an object is in free fall?
Distance vs.
Time
Graph
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Velocity vs.
Time
Graph
0 1 2 3 4 5
Acceleration vs.
Time
Graph
0 1 2 3 4 5
Lab: Musical Inclined Planes
Using the inclined plains we created for the previous lab, apply your understanding of distance versus time relationship to turn your inclined plains into musical instruments. Place your triggers for the notes along the incline plain so the timing of the rolling ball will play your song.
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Task Divisions:
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place sticky notes along the incline plain such that the ball/car will tweak them as it passes (should make a tap sound). Space these sticky notes such that the passing ball creates an even tempo. (This will help you visualize the tempo of your song along the incline plain)
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design trigger mechanism that will play a note (strike a pipe, drop a marble on a piano key, pluck a string, drop from one incline plain to another, etc.) should be made with found materials. NO DAMAGING INSTRUMENTS!
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align triggers along incline plain to create song
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LEARNING TARGETS
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apply understanding of linear motion