Distance, Location, Speed

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Topics

• Distance, Location, Speed
• Speed and Direction
• Directional quantities
• Acceleration
• Free Fall
• Graphs of Motion
• Derivatives and Integrals

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Average Speed

• distance total path length
• speed rate of travel (e.g. 50 mph)
• Average Speed distance/time (e.g. 100m in 3.0s)

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Displacement Change in Position
SI Unit meters (m)
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Velocity (m/s)
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Velocity Examples

• average velocity 60mph toward Dallas
• instantaneous velocity 1147am Northbound,
  83mph

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Example Average Velocity to 0.0s, xo 5.0m,
vo 2.0m/s t 1.2s, x 3.08m, v -5.2m/s
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Note that velocities always have directional
information. Here the - sign means x direction.
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Scalars Vectors

• Scalar size only
• e.g. speed, distance, time
• Vector magnitude and direction
• e.g. displacement, velocity, acceleration

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A honeybee travels 2 km round trip before
returning. Is the displacement for the trip the
same as the distance traveled?

1. Yes
2. No

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Acceleration (m/s/s)
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Example Car goes from 10m/s to 15m/s in a time
of 2.0 seconds. Calculate the average
acceleration.
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Previous Example to 0.0s, xo 5.0m, vo
2.0m/s t 1.2s, x 3.08m, v -5.2m/s
0
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Motion Diagrams
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• velocity arrow and position
• zero velocity is a dot
• acceleration net-force directions parallel to
  Dv
• Example slowing, reversing direction

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Kinematic Equations of Constant Acceleration
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Displacement and x vs. t Graph
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x vs. t Graph

• slope is velocity

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v vs. t Graph

• slope is acceleration

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Human Acceleration
In the 1988 Olympics, Carl Lewis reached the 20m
mark in 2.96s. Calculate average acceleration.
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Cheetah Acceleration
A cheetah can accelerate from 0 to 20m/s in 2.0s.
What is the average acceleration?
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Ex V2 Equation
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Approximate Stopping Accelerations in m/s/s Dry
Road 9 (anti-lock) 7 (skidding) Wet
Road 4 (anti-lock) 2 (skidding)
At 60mph 27m/s, what is the stopping distance
of a skid on a wet road?
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Free-Fall

• only gravity acts
• air-friction is negligible
• a 9.8m/s/s downward

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Calculus of Linear Motion

• derivatives and integrals
• Examples
• dx/dt v dv/dt a
• d/dt(3 4t 5t2) 4 10t
• v integral of acceleration

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Velocity
Example
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Summary

• speed rate of travel
• average speed distance/time.
• displacement change in position
• velocity rate position changes
• acceleration rate velocity changes
• kinematic equation set
• free fall constant acceleration.
• graphs and slopes
• derivatives and integrals of polynomials

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Example A solid metal ball is projected directly
upward with velocity 5.0m/s. How high does it
go? How long does it take to return to same
height?
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Case Study 100 meter track-race
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1. a const., 0-60 m
2. top speed of 16 m/s at 60 m.
3. a 0, 60-100 m

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100m Race
a) Acceleration and Time
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100m Race
b) Time and Distance Last 40meters of race at
constant speed of 16m/s.
Race Time tI tII 7.5s 2.5s 10.0s
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c) We can also use time found in part (a) in
velocity equation to get the acceleration of the
runner in 1st part of the race.
v vo at. 16 0 a(7.5) a 16/7.5 2.13
m/s2.
d) Distance using vavg
Dx vavgt (vo v)/2t (0 16)/2)(7.5)
(8)(7.5) 60m.
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Using v(t) equation
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Example An object has velocity of 2.0m/s at x
5.0m and at t 0.0s. At t 1.2s it has velocity
of -5.2m/s and position x 3.08m.
Average Acceleration
Consistent answer
How long did it take the object to reach v 0?
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A train moves along a straight track. The graph
shows the position as a function of time for this
train. Note that the speed at an instant is the
slope of the line at any point on the line. The
graph shows that the train
position
time

1. speeds up all the time.
2. slows down all the time.
3. speeds up part of the time and slowsdown part of
   the time.
4. moves at a constant velocity.

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Motion Diagram Example
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A car travels West at 20m/s. It begins to slow.
Use the convention that East is x. The
acceleration of the car is considered positive
since if it slowed to 19m/s in 1.0s, then
Motion Diagram
v(t)
Dv
a

-
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Net Force, Acceleration, Motion Diagrams
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Example A car starts from rest and travels West
with uniformly increasing speed. Use the
convention that East is x. Is the acceleration
or -? Is the total force acting on the car or
-? Draw a motion diagram.
Assume it goes from 0 to -10m/s in 10s.
Net-force parallel to acceleration, i.e. force is
direction.
motion diagram
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Example using Acceleration
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A car can accelerate at 6m/s/s. The time to go
from 40mph to 60mph is
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Vehicle Average Stopping Distance at 55 mph (includes reaction time)
Passenger car 190 ft.
Tractor-trailer (loaded) with cool brakes 256 ft.
Tractor-trailer (loaded) with hot brakes 430 ft.
Tractor-trailer (empty) 249 ft.
Tractor only (bobtail) 243 ft.
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Vehicle Stopping Distancefrom 60 mi/hr Stopping Distancefrom 60 mi/hr Accel. Accel.
feet meters ft/s2 m/s2
BMW M3 120 37 32.3 9.8




Dodge Colt GL 167 51 23.2 7.1
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Time to Stop
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BMW
Colt
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y and v graphs for tossed object in free-fall
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Realistic Car?
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Determine how realistic 6m/s/s is for a car by
computing the 0 to 60mph time
Good time, but can be done.