One Dimensional Motion

Honors Physics

Fall, 2014

Vector quantities

• Anything with MAGNITUDE and DIRECTION is termed a vector quantity– Scalar quantities just

have magnitude

Gravity propels a skier down a snow-covered slope at an acceleration approximately constant. The equations of “KINEMATICS”, as studied in this chapter, can give his position and velocity at any given time

Position, Distance, and Displacement

• Coordinate system defines position

• Distance total length of travel– (SI unit = meter, m)

– Scalar quantity

• Displacement change in position

Position, Distance, and Displacement

Before describing motion, you must set up a coordinate system – define an origin and a positive direction.

The distance is the total length of travel; if you drive from your house to the grocery store and back, what is the total distance you traveled?

Displacement is the change in position. If you drive from your house to the grocery store and then to your friend’s house, what is your total distance?

What is your displacement?

Average speed and velocity

• Average speed = distance / time

• Average velocity displacement divided by the total elapsed time

if

ifav tt

xx

t

xv

timeelapsed

ntdisplaceme velocity Average

Average speed and velocityWhat’s his average velocity if he returns to his starting point?

What is his average velocity if he sprints 50 m in 8 s?

What’s his average velocity if he walks back to the starting line in 40 s?

Displacement

Time taken

Displacement and Velocity

in One Dimension

Graphical Interpretation of Velocity

• The left graph shows a car moving at constant velocity (linear). The graph on the right shows a car with changing velocity. The average velocity for a given time interval is the slope of the line connecting the two coordinates in question.

Graphical Interpretation of Average Graphical Interpretation of Average Velocity Velocity

• The same motion, plotted one-dimensionally and as an position vs. time (x-t) graph:

Position vs time graphs give us information about:• average velocity slope of a line on a x-t plot is equal to the average velocity over that interval

12

12

run

rise slope

xx

yy

Graphical Interpretation of average velocity

What’s the average velocity between the intervals t = 0 s t = 3 s? Is the velocity positive or negative?

What’s the average velocity between the intervals t = 2 s t = 3 s? Is the velocity positive or negative?

• Which object is moving with constant POSITIVE velocity? Which is moving with NEGATIVE velocity? Which isn’t moving?

• HOW DO YOU KNOW?

Instantaneous VelocityInstantaneous Velocity• Instantaneous velocity

• This means that we evaluate the average velocity over a shorter and shorter period of time

Instantaneous velocity• If we have a more complex motion…

• This plot shows the average velocity being measured over shorter and shorter intervals. The instantaneous velocity is the SLOPE OF THE

LINE tangenttangent to the curve.

Instantaneous velocity

Average velocity is the slope of the straight line connecting two points corresponding to a given time interval

Instantaneous velocity is the slope of the tangent line at a given instant of time

Instantaneous velocityIs theInstantaneousvelocity at t = 0.5 s

A.Greater thanB.Less thanC.Or equal to

the instantaneous velocity at t =1.0 s?

How do you know?

Mechanics Lecture 1, Slide 18

A) YESB) NO

Are the plots shown at the left correctly related

Displacement and Velocity in One Dimension

CAN YOU EXPLAIN WHY?!?!

THERE’S ROOM OVER THERE YOU KNOW…

The velocity vs. time plot of some object is shown to the right. Which diagram below could be the Displacement vs. time plot for the same object?

A B C

Acceleration• Average acceleration the change in

velocity divided by the time it took to change the velocity

Acceleration• On Earth, gravitational

acceleration equals about 10 m/s/s

• What does it mean to have an acceleration of 10 m/s2 ?

Time (s) Velocity (m/s)

0 0

1 10

2 20

3 30

Graphical interpretation of Graphical interpretation of AccelerationAcceleration

Important Point Regarding Acceleration

• When an object’s velocity and acceleration (both vector quantities) occur in the same directionsame direction,

the object is SPEEDING UP!!!!SPEEDING UP!!!!• When an object’s velocity and acceleration

occur in opposing directionsopposing directions, the object is

SLOWING DOWN!!!SLOWING DOWN!!!– Deceleration is used to refer to a decrease in speed – Don’t confuse “negative acceleration” with

deceleration…PLEASE!!

Instantaneous AccelerationInstantaneous Acceleration

• Very similar to “instantaneous velocity” from our position vs. time graphs

• a = lim Δv / Δt t0

• The closer Δt gets to zero, the closer our ratio gets to a fixed number.

Acceleration

• Velocity vs time (v-t) graphs give us information about: average acceleration, instantaneous acceleration

• the “+” 0.25 m/s2 means the particle’s speed is increasing by 0.25 m/s every second

For the Displacement and Velocity curves shown on the left, which is the correct plot of acceleration vs. time?

AA

BB

1-D Motion with constant acceleration

• When an object moves at constant acceleration, the instantaneous acceleration at any point in a time interval equals the average acceleration divided by the whole time interval

• In other words, with constant acceleration:– Average acceleration is no

different than instantaneous acceleration!

Motion with constant Motion with constant accelerationacceleration

• If the acceleration is constant, the velocity changes linearly:

• Average velocity:• v0 = initial velocity• a = acceleration• t = time

Constant AccelerationConstant Acceleration

constanta(t) = a

Finding Displacement on a graph not using the origin as initial

coordinates

• What this graph shows is the geometrical interpretation of the following equation! Cool, huh?!

Freely Falling Objects• All objects fall towards

earth at the same constant acceleration

• Assuming air resistance is zero, of course!!

• Any object moving upward, downward, or released from rest under the influence of gravity is considered “free falling”

• A football in the air, skydiver, falling book, etc.

Motion with constant accelerationMotion with constant acceleration

An object falling in air is subject to air resistance (and therefore is not freely falling).

• Free fall is the motion of an object subject only to the influences of gravity…in most cases we’ll consider, air resistance is negligible and can be ignored.

Free falling objects

Free fall from rest: You could use thekinematics equationsto see where the numbers are comingfrom, fyi!!!

Trajectory of a projectile