Motion in One Dimension Velocity Mr. Whitney Chapter 2.1 – 2.3 in Knight

Kinematics Kinematics is the mathematical description of motion

Time and Displacement Displacement = change in position x = x f – x i y = y f – y i Change in distance = final distance – initial distance t = t f – t i Change in time= final time– initial time

Motion Table pg 31

Motion Diagram pg 31

Graph: Position vs Time pg 32

Representing 1D motion pg 32 Motion horizontally Motion vertically

1D Velocity HorizontalVertical

Compare to Slope Average Velocity v x avg =

Uniform Motion Be Careful !!! This equation only works for Uniform Motion (i.e. constant velocity) Good for when the initial position ≠ zero Alternate form: x f – x i = v x (t f – t i )

Position, Velocity, Acceleration

Component vectors All Vectors can be separated into x- vectors and y-vectors

Airplane An airplane is flying North at 500 mph when a wind of 20 mph comes up out of the East. What is the new speed and direction of the plane if no correction is made for the wind? A mph and bearing ° B mph and bearing ° C mph and bearing °

An object is pulled up a ramp at an angle of 17° with the ground level. If the force required to move this object up the incline at a constant speed is 200 pounds, how much does the object weigh? A. 654 pounds B. 209 pounds C pounds D. 684 pounds

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Planes Two planes take off from the same airport at the same time. One plane has an air speed of 500 mph and a bearing of 280°, and one plane has an air speed of 600 mph and a bearing of 35°. After 2 hours, how far apart are the planes? A miles B miles C miles

A commuter drives 15.0km on the highway at a speed of 25.0m/s, parks at work and walks 150m at a speed of 1.50m/s from his car to his office. Example b) Determine the average speed of the entire commute (a) Determine the total time of the commute.

Example: A woman starts at the entrance to a mall and walks inside for 185m north for 10minutes. She then walks 59m south in 3minutes to another store. She then leaves the store and moves south 155m in 8minutes to reach her car outside. Determine her average velocity during the trip.

Instantaneous Velocity The instantaneous speed or velocity is how fast an object is moving at a single point in time. Does the gauge on your dashboard give you speed or velocity? Does this gauge give you an average or instantaneous value?

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Graphical Analysis of Motion Position-time graph: Describes the position of object during a given time period.

Slope of x vs t graph Recall that slope = Δy / Δx

Describe the position of the objects (A-D) over time. Use origin in your statement. A B t x C D 0 EASTEAST WESTWEST What does the intersection of A and B refer to?

Describe the velocity of the objects (A-D) over time. A B t x C D 0 EASTEAST WESTWEST

Example What was the total distance traveled? What was the displacement for the entire trip? What was the average speed for the first 6 sec? What was the average velocity from B to E? What was the velocity of the object btw 2-4 sec? In which section(s) was there a constant + velocity? In which section(s) was there a constant negative velocity? In which section had the maximum speed?

v-t graphs – part 2

Graphical Analysis of Motion (2) velocity-time graph: Describes the velocity of object during a given time period.

VELOCITYVELOCITY time 0 Describe the velocity of each object during its motion, including initial velocity A B C D *Crossing t-axis = ?Intersection of lines on vt graph means ?

Instantaneous velocity Unlike v avg, instantaneous velocity occurs at a single point. How would we find v inst at t = 3.0s?

At what time(s) does the cart have a zero velocity? Describe the velocity btw s? Describe the velocity btw s?

Example Determine the displacement of the object from 20s-38s.

vt graphs – part 3

Example a) Determine the time(s) where object had - acceleration b) Determine the time(s) where object had positive non-zero velocity c) Determine the time(s) where object was at rest d) Determine the time(s) where object had constant velocity.

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Acceleration Acceleration is a changing Velocity (either magnitude and/or direction) Change in velocity Change in time Units?

Constant Acceleration Constant acceleration implies what about velocity? Constant acceleration or deceleration implies what about distance? Acceleration of zero implies what about the velocity?

Negative acceleration vs Positive acceleration: Both can equate to slowing down. When sign of acceleration matches sign of velocity, object speeds up in direction of that sign. When signs oppose, object slows down in direction of ‘v’.

Slope Interpretation If vertical is velocity? If vertical is acceleration?

Velocity equation For an object with constant acceleration

Example Determine acceleration of object between 4-9s At what time(s) did object turn around? During what time period(s) did object slow down? When did object reach maximum speed? When did object possess maximum + acceleration?

Instantaneous acceleration Instantaneous acceleration occurs at a single point. To find a inst at t = 0.6s…

Constant / Uniform Acceleration Equations

Velocity equation For an object with constant acceleration

Position Equation For an object with constant acceleration

Position equation A feather is dropped on the moon from a height of 1.40 meters. The acceleration of gravity on the moon is 1.67 m/s2. Determine the time for the feather to fall to the surface of the moon. Problem 5: 6/Sample-Problems-and-Solutions

Position equation A race car accelerates uniformly from 18.5 m/s to 46.1 m/s in 2.47 seconds. Determine the acceleration of the car and the distance traveled. Problem 4: 6/Sample-Problems-and-Solutions

Relating Velocity and Displacement For an object with constant acceleration

Relating Velocity and Displacement A bullet is moving at a speed of 367 m/s when it embeds into a lump of moist clay. The bullet penetrates for a distance of m. Determine the acceleration of the bullet while moving into the clay. (Assume a uniform acceleration.) Problem 15; Problems-and-Solutions

Relating Velocity and Displacement An engineer is designing the runway for an airport. Of the planes that will use the airport, the lowest acceleration rate is likely to be 3 m/s^2. The takeoff speed for this plane will be 65 m/s. Assuming this minimum acceleration, what is the minimum allowed length for the runway? Problem 8; 6/Sample-Problems-and-Solutions

Relating Velocity and Displacement A baseball is popped straight up into the air and has a hang-time of 6.25 s. Determine the height to which the ball rises before it reaches its peak. (Hint: the time to rise to the peak is one-half the total hang-time.) Problem 13; Problems-and-Solutions

EXAMPLE While driving along at 20m/s, you notice the light up ahead turns red (110m away). Assuming you have a reaction time of 0.5s, a) How far from the light are you when you begin to apply the brakes? b) What constant acceleration will bring you to rest at the light?

EXAMPLE 2 A car starts from rest at a stop sign. It accelerates uniformly at 4.0m/s 2 for 6.0s, coasts for 2.0s, and then slows down at 3.0m/s 2 for the next stop sign. a) How far apart are the stop signs? b) Determine the maximum velocity during the trip.

a vs t graph a t 0 We will only deal with constant accelerations.

Reference Frames & Relative Motion Any measurement of position, distance, or speed must be made

In order to determine the speed of object moving in a particular RF, we use subscripts

V SG =20m/s V BG =6m/s V CG = -30m/s How fast is bike moving relative to bus? How fast is bus moving relative to car?

Example: A railroad flatcar is traveling to the right at a speed of 13.0 m/s relative to an observer standing on the ground. Someone is riding a motor scooter on the flatcar. Determine the velocity of the motor scooter relative to the flatcar if its velocity relative to the observer on the ground is 3.0 m/s to the left?

Falling & Acceleration

FREEFALL

Anatomy of a upwardly thrown object

A ball is thrown upward with an initial speed of 15.0m/s Assume negligible air resistance. EXAMPLE 1 a) Find the maximum height attained by the ball. b) How much time does it take to reach the apex? c) Determine the velocity 2.2s into flight.

Example: Calculate the impact speed with the water after falling from 40ft above the water, from rest. Assuming he stops in a depth of 12inches of water, determine his acceleration, assuming it to be constant.

EXAMPLE 2 As a part of a movie stunt a stunt man hangs from the bottom of an elevator that is rising at a steady rate of 1.10m/s. The man lets go of the elevator and freefalls for 1.50s before being caught by the end of a rope that is attached to the bottom of the elevator. (a) Calculate the velocity of the man at the instant he is caught by the rope. (b) How long is the rope?

EXAMPLE 3 An honors physics student stands at the edge of a cliff that is 36m high. He throws a water balloon straight up at 12.5m/s so that it just misses the edge of the cliff on the way down. Determine velocity of balloon as it strikes ground below (many ways to solve)

Three students are standing side-by-side next to the railing on a fifth floor balcony. Simultaneously, the three students release their pennies. One student drops a penny to the ground below. The second student tosses penny straight downwards at 15 m/s while third student tosses penny straight upwards at 15 m/s. Assume freefall. d) Which penny or pennies strike(s) the ground with the greatest acceleration? a) Which penny or pennies strike(s) ground first? b) Which penny or pennies strike(s) ground last? c) Which penny or pennies strike(s) the ground with the greatest final velocity? Collaborate with person next to you to answer following questions: