Motions under non-constant acceleration

How do we know the position from the velocity function v(t)?

Displacement from a Velocity - Time Graph

The displacement of a particle during the time interval ti to tf is equal to the area under the curve between the initial and final points on a velocity-time curve

Displacement during a time interval

First, we divide the time range from Ta to Tf into N equal intervals

N

TTt af

• Displacement xn during the n-th interval

tvxt

xv nn

nn

Total displacement from Ta to Tf

N

nn

N

nn tvxx

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In the limit as N approaches to infinite, the time interval t becomes smaller and smaller.

f

a

T

T

N

nn

t

dttvtvx )(10

lim

Displacement is the area under the curve of v(t).

Position xf at Tf with initial position xa at Ta

f

a

T

Taf dttvxx )(

If the function v(t) is constant in time,

)( afaf TTvxx

How do we know the velocity from the acceleration function a(t) ?

Velocity vf at Tf with initial velocity va at Ta

f

a

T

Taf dttavv )(

If the function a(t) is constant in time,

)( afaf TTavv

Position xf at Tf under a constant acceleration with initial velocity va at Ta=0

atvtv a )(

2

0

2

1

)(

ffaa

T

aaf

aTTvx

dtatvxxf

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Chapter 3

Motion in Two Dimensions

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3.1 Position and Displacement The position of an

object is described by its position vector,

The displacement of the object is defined as the change in its position

Fig 3.1

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Average Velocity The average velocity is

the ratio of the displacement to the time interval for the displacement

The direction of the average velocity is the direction of the displacement vector,

Fig 3.2

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Average Velocity, cont The average velocity between points is

independent of the path taken This is because it is dependent on the

displacement, which is also independent of the path

If a particle starts its motion at some point and returns to this point via any path, its average velocity is zero for this trip since its displacement is zero

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Instantaneous Velocity The instantaneous velocity is the limit of

the average velocity as ∆t approaches zero

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Instantaneous Velocity, cont The direction of the instantaneous

velocity vector at any point in a particle’s path is along a line tangent to the path at that point and in the direction of motion

The magnitude of the instantaneous velocity vector is the speed

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Average Acceleration The average acceleration of a particle

as it moves is defined as the change in the instantaneous velocity vector divided by the time interval during which that change occurs.

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Average Acceleration, cont

As a particle moves, can be found in different ways

The average acceleration is a vector quantity directed along

v

Fig 3.3

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Instantaneous Acceleration The instantaneous acceleration is the

limit of the average acceleration as approaches zero

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Producing An Acceleration Various changes in a particle’s motion

may produce an acceleration The magnitude of the velocity vector may

change The direction of the velocity vector may

change Even if the magnitude remains constant

Both may change simultaneously

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3.2 Kinematic Equations for Two-Dimensional Motion When the two-dimensional motion has a

constant acceleration, a series of equations can be developed that describe the motion

These equations will be a generlization of those of one-dimensional kinematics to the vector form.

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Kinematic Equations, 2 Position vector

Velocity

Since acceleration is constant, we can also find an expression for the velocity as a function of time:

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Kinematic Equations, 3

The velocity vector can be represented by its components

is generally not along the direction of either or

Fig 3.4(a)

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Kinematic Equations, 4 The position vector can also be

expressed as a function of time:

This indicates that the position vector is the sum of three other vectors:

The initial position vector The displacement resulting from The displacement resulting from

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Kinematic Equations, 5

The vector representation of the position vector

is generally not in the same direction as or as

and are generally not in the same direction

Fig 3.4(b)

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Kinematic Equations, Components The equations for final velocity and final

position are vector equations, therefore they may also be written in component form

This shows that two-dimensional motion at constant acceleration is equivalent to two independent motions One motion in the x-direction and the other

in the y-direction

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Kinematic Equations, Component Equations becomes

vxf = vxi + axt and vyf = vyi + ayt

becomes

xf = xi + vxi t + 1/2 axt2 and yf = yi + vyi t + 1/2 ayt2

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30

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3.3 Projectile Motion The motion of an object under the

influence of gravity only The form of two-dimensional motion

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Assumptions of Projectile Motion The free-fall acceleration is constant

over the range of motion And is directed downward

The effect of air friction is negligible With these assumptions, the motion of

the object will follow

33

Projectile Motion Vectors

The final position is the vector sum of the initial position, the displacement resulting from the initial velocity and that resulting from the acceleration

This path of the object is called the trajectory

Fig 3.6

34

Analyzing Projectile Motion Consider the motion as the

superposition of the motions in the x- and y-directions

Constant-velocity motion in the x direction ax = 0

A free-fall motion in the y direction ay = -g

35

Verifying the Parabolic Trajectory Reference frame chosen

y is vertical with upward positive Acceleration components

ay = -g and ax = 0

Initial velocity components vxi = vi cos i and vyi = vi sin i

36

Projectile Motion – Velocity at any instant The velocity components for the

projectile at any time t are: vxf = vxi = vi cos i = constant

vyf = vyi – g t = vi sin i – g t

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Projectile Motion – Position Displacements

xf = vxi t = (vi cos i t yf = vyi t + 1/2ay t2 = (vi sinit - 1/2 gt2

Combining the equations gives:

This is in the form of y = ax – bx2 which is the standard form of a parabola

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What are the range and the maximum height of a projectile

The range, R, is the maximum horizontal distance of the projectile

The maximum height, h, is the vertical distance above the initial position that the projectile can reaches.

Fig 3.7

39

Projectile Motion Diagram

Fig 3.5

40

Projectile Motion – Implications The y-component of the velocity is zero

at the maximum height of the trajectory The accleration stays the same

throughout the trajectory

41

Height of a Projectile, equation The maximum

height of the projectile can be found in terms of the initial velocity vector:

The time to reach the maximum:

sin

gi i

m

vt

42

Range of a Projectile, equation The range of a projectile

can be expressed in terms of the initial velocity vector:

The time of flight = 2tm

This is valid only for symmetric trajectory

43

More About the Range of a Projectile

44

Range of a Projectile, final

The maximum range occurs at i = 45o

Complementary angles will produce the same range The maximum height will be different for

the two angles The times of the flight will be different for

the two angles

45

Non-Symmetric Projectile Motion

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48

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50

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Fig 3.10

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58

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