# Kinematics in One Dimension

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Kinematics in One Dimension
Describing Motion: Kinematics in One Dimension

Sign Convention & Direction

Distance & Displacement
Distance (x) equates Displacement equates to

Displacement Displacement is written:

Example A person moves on the number line shown below. The person begins at B, walks to C, and then turns around and walks to A. For this entire range of motion DETERMINE: the person’s final position the displacement the distance. m m m A B C

Speed & Velocity Speed: How far Velocity: How

Average Speed & Velocity

Example 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. (a) Determine the total time of the commute. b) Determine the average speed of the entire commute

EXAMPLE Usain Bolt holds the record for the 100m sprint completing it in only 9.58s! a) Determine his average speed in m/s. (1.6km = 1mi) Did he run faster than this at some point? b) Mr Sample (I hold no record) ran the Philly half-marathon (13.1mi) in 1hr55min36sec. Determine my avg speed in mph.

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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Acceleration Acceleration is 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? 14

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’.

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

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

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

Slope Interpretation

Describe the velocity of the objects (A-D) over time.
x A E A S T B t WE S T C D

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 velocity of the object btw 2-4 sec? What was the average velocity from B to E? 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?

Instantaneous velocity
Unlike vavg, instantaneous velocity occurs at a single point. How would we find vinst 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?

a) During what time periods, if any, is the object's velocity constant?
b) At what time is its velocity the greatest? c) At what time, if any, is the velocity zero? d) Does the object run in one direction or in both along its tunnel during the time shown?

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

Describe the velocity of each object during its motion, including initial velocity
V E L O C I T Y A B time C D Intersection of lines on vt graph means ? *Crossing t-axis = ?

Slope of v vs t graph

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.

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 ainst at t = 0.6s… 30

Constant Acceleration Eqns
We can write avg velocity 2 different ways: Combining the two eqns yields:

Constant / Uniform Acceleration Equations

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/s2 for 6.0s, coasts for 2.0s, and then slows down at 3.0m/s2 for the next stop sign. a) How far apart are the stop signs? b) Determine the maximum velocity during the trip.

v-t graphs – part 2

v-t graphs – part 2 Determine the displacement of the object from 20s-38s.

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

VBG=6m/s VSG =20m/s VSG =20m/s VCG= -30m/s
How fast is bike moving relative to bus? VSG =20m/s VCG= -30m/s How fast is bus moving relative to car? 40

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Falling & Acceleration
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FREEFALL 43 43

Anatomy of a upwardly thrown object

EXAMPLE 1 A ball is thrown upward with an initial speed of 15.0m/s Assume negligible air resistance.  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 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.   a) m/s b) 11m - find disp of man + dist of elevator in 1.5s (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) 47

Collaborate with person next to you to answer following questions:
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. 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? d) Which penny or pennies strike(s) the ground with the greatest acceleration?