# Curvilinear Motion and Projectiles

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8/12/2019 Curvilinear Motion and Projectiles

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Curvilinear Motion & Projectiles

Learning Objectives:

To be able to:

1. Describe the motion of a particle traveling along a

curved path.2. Relate kinematic quantities in terms of the rectangular

components of the vectors.

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APPLICATIONS

The path of motion of a plane canbe tracked with radar and its x, y,

and z coordinates (relative to a

point on earth) recorded as a

function of time.

How can we determine the velocity

or acceleration of the plane at any

instant?

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GENERAL CURVILINEAR MOTION

(Section 12.4)

A particle moving along a curved path undergoes curvilinear motion.

Since the motion is often three-dimensional, vectorsare used to

describe the motion.

Thepositionof the particle at any instant is designated by the vector

r= r(t). Both the magnitudeand directionof rmay vary with time.

A particle moves along a curve

defined by the path function, s.

If the particle moves a distance Ds

along the curve during time interval

Dt, the displacementis determined by

vectorsubtraction: D r= r- r

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VELOCITY

Velocityrepresents the rate of change in the position of a

particle.The averagevelocityof the particle

during the time increment Dt is

vavg= Dr/Dt .

The instantaneous velocityis thetime-derivative of position

v= dr/dt .

The velocity vector, v, is always

tangent to the path of motion.The magnitude of vis called the speed. Since the arc length Ds

approaches the magnitude of Dras t0, the speed can be

obtained by differentiating the path function (v = ds/dt).

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ACCELERATION

Accelerationrepresents the rate of change in the

velocity of a particle.

If a particles velocity changes from vto vover a

time increment Dt, the average accelerationduring

that increment is:

aavg= Dv/Dt = (v- v)/Dt

The instantaneous accelerationis the time-

derivative of velocity:

a= dv/dt = d2r/dt2

A plot of the locus of points defined by the arrowhead

of the velocity vector is called a hodograph. The

acceleration vector is tangent to the hodograph, but

not, in general, tangent to the path function.

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CURVILINEAR MOTION: RECTANGULAR COMPONENTS

(Section 12.5)

It is often convenient to describe the motion of a particle in

terms of its x, y, z or rectangularcomponents, relative to a fixed

frame of reference.

The magnitudeof the position vector is: r = (x2+ y2+ z2)0.5

The directionof r is defined by the unit vector: ur= (1/r)r

The position of the particle can be

defined at any instant by the

position vectorr = x i + yj + z k .

The x, y, z components may all be

functions of time, i.e.,

x = x(t), y = y(t), and z = z(t) .

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RECTANGULAR COMPONENTS: VELOCITY

The magnitudeof the velocity

vector is

v = [(vx)2+ (vy)2+ (vz)2]0.5

The directionof vis tangent

to the path of motion.

The velocity vectoris the time derivative of the position vector:

v= dr/dt = d(xi)/dt + d(yj)/dt + d(zk)/dt

Since the unit vectorsi,j, kare constantin magnitudeand

direction, this equation reduces to v = vx i+ vyj+ vz k

where vx= = dx/dt, vy= = dy/dt, vz= = dz/dtx y z

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RECTANGULAR COMPONENTS: ACCELERATION

The directionof ais usually

not tangentto the path of the

particle.

The acceleration vectoris the time derivative of the

velocity vector (second derivative of the position vector):

a= dv/dt = d2r/dt2= ax i+ ayj+ az k

where ax= = = dvx /dt, ay= = = dvy /dt,

az= = = dvz /dt

vx x vy y

vz z

The magnitudeof the acceleration vector is

a = [(ax)2+ (ay)

2+ (az)2]0.5

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Application Note

Often, we can get dynamics problems where we have y=f(x) and we haveVX = f(t), however, we cannot easily get VY= f(t). In this case, we can

apply the chain rule from calculus:

dt

dx

dx

dy

dt

dv

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MOTION OF A PROJECTILE

Section 12.6

Learning Objectives:

Be able to:

1. Analyze the free-flight motion of a projectile.

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APPLICATIONS

A good kicker instinctively knows at what angle, q, and initial velocity, vA,

he must kick the ball to make a field goal.

For a given kick strength, at what angle should the ball be kicked to getthe maximum distance?

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MOTION OF A PROJECTILE (Section 12.6)

Projectile motion can be treated as two rectilinear

motions, one in the horizontal direction experiencing

zero accelerationand the other in the vertical

direction experiencing constant acceleration(i.e.,from gravity).

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MOTION OF A PROJECTILE (Section 12.6)

For illustration, consider the two balls on

the left. The red ball falls from rest,

whereas the yellow ball is given a

horizontal velocity. Each picture in this

sequence is taken after the same time

interval. Notice both balls are subjected tothe same downward acceleration since

they remain at the same elevation at any

instant. Also, note that the horizontal

distance between successive photos of theyellow ball is constant since the velocity

in the horizontal direction is constant.

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KINEMATIC EQUATIONS: HORIZONTAL MOTION

Since ax = 0, the velocity in the horizontal direction remains constant (vx=

vox) and the position in the x direction can be determined by:

x = xo+ (vox) t

Note: We neglect wind resistance in his analysis!

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KINEMATIC EQUATIONS: VERTICAL MOTION

Since the positive y-axis is directed upward, ay=g. Application of the

constant acceleration equations yields:

vy= voyg t

y = yo+ (voy) t g t2

vy2= voy

22 g (yyo)

We get the last equation by solving the first two equations for t. There

only two of these equations are independent. You can only sove for

two unknowns.

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