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Accelerated Motion Velocity, acceleration and gravity

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How fast do things fall

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Reflexes

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Position-Time Graphs 1234

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Velocity v. Time

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Definitions Velocity Velocity Change in position with respect to time Change in position with respect to time v = Δd/Δt v = Δd/Δt Which can be written as: (d final -d initial )/(t final -t initial ) Common notation: (d f – d i )/(t f –t i ) Acceleration Change in velocity with respect to time a = Δv/Δt Which can be written as: (v final -v initial )/(t final -t initial ) Common notation : (v f –v i )/(t f –t i )

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Velocity to Acceleration v=Δd/Δt=(d final –d initial )/(t final –t initial ) v=Δd/Δt=(d final –d initial )/(t final –t initial ) a=Δv/Δt=(v final –v initial )/(t final –t initial ) a=Δv/Δt=(v final –v initial )/(t final –t initial )

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Average Acceleration a =Δv/Δt=(v final –v initial )/(t final –t initial ) a =Δv/Δt=(v final –v initial )/(t final –t initial ) f = final f = final i = initial i = initial If t initial = 0 If t initial = 0 a = (v final – v initial )/t final a = (v final – v initial )/t final Or: Or: v f = v i +at f v f = v i +at f

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Positive and Negative Acceleration

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Practice Problem A soccer player is running at a constant velocity of 50.0km/h (31mph). The player falls and skids to a halt in 4.0 seconds. A soccer player is running at a constant velocity of 50.0km/h (31mph). The player falls and skids to a halt in 4.0 seconds. What is the average acceleration of the player during the skid? What is the average acceleration of the player during the skid? What is the plot of the velocity vs. time? What is the plot of the velocity vs. time?

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Practice Problem A water balloon in the sling of a water balloon launcher undergoes a constant acceleration 25m/s^2 for 1.5s. A water balloon in the sling of a water balloon launcher undergoes a constant acceleration 25m/s^2 for 1.5s. What is the velocity of the water balloon right after launch? What is the velocity of the water balloon right after launch?

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Practice Problem A car accelerates from rest at 5 m/s2 for 5 seconds. It moves with a constant velocity for some time, and then decelerates at 5 m/s2 to come to rest. The entire journey takes 25 seconds. Plot the velocity-time graph of the motion. A car accelerates from rest at 5 m/s2 for 5 seconds. It moves with a constant velocity for some time, and then decelerates at 5 m/s2 to come to rest. The entire journey takes 25 seconds. Plot the velocity-time graph of the motion.

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Practice Problem Determine the accelerations for a 1, a 2, a 3, and a 4 for each time interval. a 1 = 4/5 a 2 = (4-4)/(10-5) a 3 = (16-4)/(20-10) a 4 = (0-16)/(30-20)

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Frictionless Cars Hypotenuse Gravity

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Frictionless Car Plots

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Good Reading on Plot

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Velocity with Constant Acceleration Given: Given: Solve Solve for tf: for tf: Substitute in: Substitute in: Yeilds: Yeilds: 2 ½ ½ 2

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Velocity with Constant Acceleration Solve for vf Solve for vf ½ 2 ½ 2 1 1 2 2 2 1 1 2 2 2 2 2 2 2 2 22 2 2 2

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Graphs Determine which equations provide the area under the graph. (let ti = 0) Determine which equations provide the area under the graph. (let ti = 0) ½ (v f -v i ) tftftftf ½ a (t f -t i ) 2 (v f -v i ) (t f -t i ) 2 1 2 Velocity (m/s) Time (s) 1) 2) 3)

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Velocity with Constant Acceleration Equation to remember: Equation to remember: v f ^2 = v i ^2 +2a(d f -d i ) v f ^2 = v i ^2 +2a(d f -d i )

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Position with Average Acceleration Δd/Δt = Δv + ½ a Δt Δd/Δt = Δv + ½ a Δt Δd = ΔvΔt + ½ a Δt^2 Δd = ΔvΔt + ½ a Δt^2 When t i = 0: When t i = 0: d f - d i = v i t f + ½ at f ^2 d f - d i = v i t f + ½ at f ^2

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Position with Average Acceleration Equation to remember: Equation to remember: Final position = initial position + (change in velocity)*time + ½ (acceleration)*(time squared) Final position = initial position + (change in velocity)*time + ½ (acceleration)*(time squared) d f = d i + (v f -v i )t + ½ at^2 d f = d i + (v f -v i )t + ½ at^2

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Table 3-3 Page 68 Equations of Motion for Uniform Acceleration Equations of Motion for Uniform Acceleration Equation Variable s Initial Conditi ons Average Acceleration t f,v f,a vivivivi Velocity with Constant Acceleration t f,d f,a d i,v i Position with Average Acceleration d f,v f, a d i,v i

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Free Fall on the Moon

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Group Project pg 78 How fast is the Earth spinning? How fast is the Earth spinning? 0.5 km/sec 0.5 km/sec How fast is the Earth revolving around the Sun? How fast is the Earth revolving around the Sun? 30 km/sec 30 km/sec How fast is the Solar System moving around the Milky Way Galaxy? How fast is the Solar System moving around the Milky Way Galaxy? 250 km/sec 250 km/sec How fast is our Milky Way Galaxy moving in the Local Group of galaxies? How fast is our Milky Way Galaxy moving in the Local Group of galaxies? 370 km/sec 370 km/sec

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Free Fall All Objects fall at the same speed regardless of mass (if you can neglect wind resistance). All Objects fall at the same speed regardless of mass (if you can neglect wind resistance).

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Free Fall A ball or a bullet? A ball or a bullet?.. Position with Average Acceleration Height with constant gravity

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Falling

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Poor little guy

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Graphs

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Cart Movement

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Practice Problems If you throw a ball straight upward, it will rise into the air and then fall back down toward the ground. If you throw a ball straight upward, it will rise into the air and then fall back down toward the ground. Imagine that you throw the ball with an initial velocity of 10.0 m/s. Imagine that you throw the ball with an initial velocity of 10.0 m/s. a. How long does it take the ball to reach the top of its motion? a. How long does it take the ball to reach the top of its motion? b. How far will the ball rise before it begins to fall? b. How far will the ball rise before it begins to fall? c. What is its average velocity during this period? c. What is its average velocity during this period?

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a. How long does it take the ball to reach the top of its motion?

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b. How far will the ball rise before it begins to fall?

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c. What is its average velocity during this period?

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Practice Problem A sudden gust of wind increases the velocity of a sailboat relative to the water surface from 3.0 m/s to 5.5 m/s over a period of 60.0 s. A sudden gust of wind increases the velocity of a sailboat relative to the water surface from 3.0 m/s to 5.5 m/s over a period of 60.0 s. a. What is the average acceleration of the sailboat? a. What is the average acceleration of the sailboat? b. How far does the sailboat travel during the period of acceleration? b. How far does the sailboat travel during the period of acceleration?

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a. What is the average acceleration of the sailboat?

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b. How far does the sailboat travel during the period of acceleration?

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Practice Problem A car starts from rest with an acceleration of 4.82 m/s^2 at the instant when a second car moving with a velocity of 44.7 m/s (100mph by the way) passes it in a parallel line. How far does the first car move before it overtakes the second car? A car starts from rest with an acceleration of 4.82 m/s^2 at the instant when a second car moving with a velocity of 44.7 m/s (100mph by the way) passes it in a parallel line. How far does the first car move before it overtakes the second car? Setup an equation or graph Setup an equation or graph

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Setup the equation 00

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Position d

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Velocity v=Δd/Δt

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Acceleration a=Δv/Δt

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