linear motion: velocity and...
TRANSCRIPT
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Linear Motion: Velocity and Acceleration
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Relative Motion • Everything moves, even things at rest • Relative – regarded in relation to something
else; depends on point of view, or frame of reference
• A book at “rest”, relative to you, is actually moving at 30 km/s with respect to the sun, and even faster the center of the galaxy
• Unless stated otherwise, when we discuss speeds of things in our environment, we mean speed with respect to the surface of Earth
• Motion is relative!
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Motion is relative!
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Speed • Speed – The measure of how fast something is
moving; the rate at which distance is covered. • Instantaneous Speed – The speed at any
instant in time • Average Speed – The total distance covered
divided by the total time
Measured in meters/second [m/s]
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Average vs. Instantaneous Speed
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Which of the following are speeds? (Hint: There may be more than one correct answer!)
a. 5 m b. 10 m/s c. 4567 mph d. 0.009 s e. 17 miles f. 0.023 km/hr g. 12 cm/min
Check for Understanding
a. 5 m b. 10 m/s c. 4567 mph d. 0.009 s e. 17 miles f. 0.023 km/hr g. 12 cm/min
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Velocity
• Velocity – Speed in a given direction ▫ Velocity can be (+) or (-)
• Constant Velocity – Requires that both constant speed and constant direction must be maintained
• Changing Velocity – Either the speed or the direction changes, so the velocity changes
Measured in meters/second [m/s]
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• Average Velocity – change in position/change in time.
• **Note: Average speed is different from average velocity. ▫ Average speed = distance traveled/time ▫ Average velocity = change in position/change in
time
€
vavg =ΔxΔt
vavg= average speed [m/s] Δ = change x = position [m] t = time [s]
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• The speedometer in every car also has an odometer that records the distance traveled. ▫ If the odometer reads zero at the beginning of a
trip and 35 km a half hour later (assuming the car is traveling forward in a straight line), what is the average velocity?
Check for Understanding
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What is the difference between speed and velocity?
Speed is always (+). Speed does not have direction.
Velocity can be (+) or (-). Velocity has direction.
Speed = 5 m/s Velocity = -5 m/s
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Acceleration • Acceleration – The rate at which velocity is
changing • Term applies to both increases and decreases in
velocity (difference between positive and negative acceleration)
Measured in meters/second² (m/s²)
aavg= average acceleration[m/s2] Δ = change v = velocity[m/s] t = time [s]
€
aavg =ΔvΔt
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• Acceleration applies to changes in direction also; when the direction changes, the acceleration changes
• Most of the time we will concern ourselves with motion in a straight line, and can look at the change in speed.
• Gravity – The acceleration that causes objects to move towards the Earth or other large objects
Gravity = g = 9.8 m/s² ≈ 10 m/s2
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• In 1977 off the coast of Australia, the fastest speed by a vessel on the water was achieved. If this vessel were to undergo an average acceleration of 1.80 m/s2, it would go from rest to its top speed in 85.6 s. What was the speed of this vessel?
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Velocity vs. Acceleration
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Observe the animation of the three cars below. Which car or cars (red, green, and/or blue) are undergoing an acceleration? Study each car individually in order to determine the answer.
If necessary, review the definition of acceleration.
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In Summary… • Speed and __________ are different because
_____________ is a measure of how fast something is moving, and velocity is a measure of how fast something is moving in a certain ___________.
• Average __________ is the rate at which the velocity is changing with time.
• ___________ is the change in position divided by the change in time.
• The units of acceleration are __________.
Check for Understanding
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Linear Motion: Part II Chapter 2
FLT: I can solve word problems using the equations for linear motion with constant acceleration.
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What we already know: • Velocity tells us how fast and in what direction
an object is moving. • Acceleration tells us the rate at which the
velocity is changing.
€
vavg =ΔxΔt
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aavg =ΔvΔt
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a= acceleration [m/s2] v0 = initial velocity [m/s] v = final velocity t = time [s]
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v = at + v0
€
aavg =ΔvΔt
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Example 1 • A runner is initially moving at 0.5 m/s and
accelerates at the rate of 0.9 m/s2 for 6 s. What is the velocity of the runner after the 6 s?
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How far does an object move? • The position of an object is related to the
acceleration, time, initial velocity, and initial position of the object.
€
x =12at 2 + v0t + x0
x or y = position [m] x0 or y0 = initial position [m] a = acceleration [m/s2] v0 = initial velocity [m/s] t = time [s]
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Example 2 • A ball is rolling across a table with an initial
velocity of 5 m/s. It accelerates at 2 m/s2. If the ball rolls for 3 seconds, how much distance does it cover?
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Example 3 • A ball moving with a speed of 2.00 m/s increases
speed uniformly, so that in 40 s it has traveled 70.2 m. What is the magnitude of the ball’s acceleration?
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One Last Equation… • If you are given a problem where you are not
give time, and you are not asked to find time, use:
€
v 2 − v02 = 2a(x − x0)
x = position [m] x0 = initial position [m] a = acceleration [m/s2] v0 = initial velocity [m/s] v = velocity [m/s]
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Example • A radio-controlled toy car increases speed over a
distance 0f 15.2 m. If the car starts at rest and has a final speed of 0.76 m/s, what is the magnitude of its acceleration?
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Linear Motion Equations
1.
2.
3.
4.
€
v 2 − v02 = 2a(x − x0)
€
x =12at 2 + v0t + x0€
v = at + v0
€
vavg =ΔxΔt
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Linear Motion: Free Fall Chapter 2
FLT: I can solve free fall problems using the 3 equations for linear motion with constant acceleration.
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• Free Fall – Objects that are only affected by gravity (neglecting air resistance)
• Elapsed Time – The time that has passed since the beginning of the fall
• We can find out about the motion of objects in free fall using the four linear motion equations.
Free Fall
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Linear Motion Equations
1.
2.
3.
4.
€
v 2 − v02 = 2a(x − x0)
€
x =12at 2 + v0t + x0€
v = at + v0
€
vavg =ΔxΔt
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Acceleration due to gravity, g
• If an object is in free fall (aka falling through the air towards the earth), it ALWAYS accelerates at ‘g’. (Remember, g = 10 m/s2)
• Gravity pulls objects towards earth (down), therefore we use acceleration due to gravity a = -g when we are solving free fall problems
• In other words, a = -10 m/s2 for ANY object in free fall – ALWAYS!
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• Miss Stein drops a ball off the roof of the gym. What is its acceleration?
• Miss Stein throws a tomato into the air. What is its acceleration?
Check for Understanding
-10 m/s2
-10 m/s2
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Keys for solving free fall problems: 1. Draw a picture! 2. a = -10 m/s2
For objects moving up then down (as below) 1. v at top = 0 m/s 2. t to top = 1/2 (total t)
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Example 2 • A pumpkin is released from rest at the top of the
gym, which is 271 m tall. Disregarding air resistance, calculate the displacement of the pumpkin after 2 s.
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Example • Miss Stein throws an orange straight up into the
air with a speed of 60 m/s. The orange is in the air for 12 seconds. How high does the orange rise?
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Air Resistance
• Air resistance is responsible for the differences in accelerations that we see between an elephant and a feather. • With a lack of air, these two items would fall at
the same rate!! • Air resistance less noticeably effects more
condense objects (i.e. baseballs and stones)
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The Elephant and the Feather, Air Drag