Kinematics Chapter 2 - Siena Sciencerfinn/phys130/P130-3-1Dmotion.pdfChapter 2 Motion in One Dimension Kinematics ... velocity vs. time graph is the acceleration ... displacement –

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Chapter 2

Motion in One Dimension

Kinematics

• Describes motion while ignoring theagents that caused the motion

• For now, will consider motion in onedimension• Along a straight line

• Will use the particle model• A particle is a point-like object, has mass

but infinitesimal size

Position

• Defined in terms of aframe of reference• One dimensional, so

generally the x- or y-axis

• The object’s positionis its location withrespect to the frameof reference

Position-Time Graph

• The position-timegraph shows themotion of theparticle (car)

• The smooth curve isa guess as to whathappened betweenthe data points

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Displacement

• Defined as the change in positionduring some time interval• Represented as ΔxΔx = xf - xi

• SSI units are meters (m) Δx can be positiveor negative

• Different than distance – the length of apath followed by a particle

Vectors and Scalars

• Vector quantities need both magnitude (sizeor numerical value) and direction tocompletely describe them• Ex: displacement• Will use + and – signs to indicate vector directions

• Scalar quantities are completely described bymagnitude only• Ex: distance

Average Velocity

• The average velocity is rate at whichthe displacement occurs

• The dimensions are length / time [L/T]• The SI units are m/s• Is also the slope of the line in the

position – time graph

Average Speed

• Speed is a scalar quantity• same units as velocity

• total distance / total time

• The average speed is not (necessarily)the magnitude of the average velocity

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Ch 2: Problem 4

• A particle moves according to theequation x = 10 t2 where x is in metersand t is in seconds.(a) Find the average velocity for thetime interval from 2.00 s to 3.00 s.

Instantaneous Velocity

• The limit of the average velocity as thetime interval becomes infinitesimallyshort, or as the time intervalapproaches zero

• The instantaneous velocity indicateswhat is happening at every point of time

Instantaneous Velocity,equations

• The general equation for instantaneousvelocity is

• The instantaneous velocity can bepositive, negative, or zero

Instantaneous Velocity, graph

• The instantaneousvelocity is the slopeof the line tangent tothe x vs. t curve

• This would be thegreen line

• The blue lines showthat as Δt getssmaller, theyapproach the greenline

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Instantaneous Speed

• The instantaneous speed is themagnitude of the instantaneous velocity

• Remember that the average speed isnot the magnitude of the averagevelocity

Ch 2: Problem 8

• A position-time graph for aparticle moving along the xaxis is shown.(a) Find the average velocityin the time interval t=1.50 s tot=4.00 s.(b) Determine theinstantaneous velocity att=2.00 s by measuring theslope of the tangent lineshown in the graph.(c) At what value of t is thevelocity zero?

Average Acceleration

• Acceleration is the rate of change of thevelocity

• Dimensions are L/T2

• SI units are m/s2

Instantaneous Acceleration

• The instantaneous acceleration is the limit ofthe average acceleration as Δt approaches 0

• Acceleration is curvature of position-timegraph

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Instantaneous Acceleration --graph• The slope of the

velocity vs. time graphis the acceleration

• The green linerepresents theinstantaneousacceleration

• The blue line is theaverage acceleration

Ch 2: Probem 11

• A 50.0-g superball traveling at 25.0 m/sbounces off a brick wall and reboundsat 22.0 m/s. A high-speed camerarecords this event. If the ball is incontact with the wall for 3.50 ms, whatis the magnitude of the averageacceleration during this time interval?

Acceleration and Velocity, 1

• When an object’s velocity andacceleration are in the same direction,the object is speeding up

• When an object’s velocity andacceleration are in the oppositedirection, the object is slowing down

Acceleration and Velocity, 2

• The car is moving with constant positive velocity(shown by red arrows maintaining the same size)

• Acceleration equals zero

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Acceleration and Velocity, 3

• Velocity and acceleration are in the same direction• Acceleration is uniform (blue arrows maintain the

same length)• Velocity is increasing (red arrows are getting longer)• This shows positive acceleration and positive velocity

Acceleration and Velocity, 4

• Acceleration and velocity are in opposite directions• Acceleration is uniform (blue arrows maintain the

same length)• Velocity is decreasing (red arrows are getting shorter)• Positive velocity and negative acceleration

Kinematic Equations --summary Ch 2: Problem 19

• Jules Verne in 1865 suggested sendingpeople to the Moon by firing a space capsulefrom a 220-m-long cannon with a launchspeed of 10.97 km/s. What would have beenthe unrealistically large accelerationexperienced by the space travelers duringlaunch?

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Ch 2: Problem 20

• A truck covers 40.0 m in 8.50 s whilesmoothly slowing down to a final speedof 2.80 m/s. (a) Find its original speed.(b) Find its acceleration.

Graphical Look at Motion –displacement – time curve

• The slope of thecurve is the velocity

• The curved lineindicates the velocityis changing• Therefore, there is

an acceleration

Graphical Look at Motion –velocity – time curve

• The slope gives theacceleration

• The straight lineindicates a constantacceleration

• The zero slopeindicates a constantacceleration

Graphical Look at Motion –acceleration – time curve

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Ch 2: Problem 15

• A particle moves along the x axisaccording to the equationx = 2.00 + 3.00 t - 1.00 t2

where x is in meters and t is in seconds.At t=3.00 s, find(a) the position of a particle,(b) its velocity,(c) its acceleration.