KINEMATICS OF PARTICLES

PROBLEMS ON

RELATIVE MOTION WITH RESPECT TO

TRANSLATING AXES

1. The car A has a forward speed of 18 km/h and is accelerating at 3

m/s2. Determine the velocity and acceleration of the car relative to

observer B who rides in a nonrotating chair on the Ferris wheel. The

angular rate W= 3 rev/min of the Ferris wheel is constant. (2/188)

vA = 18 km/h, aA = 3 m/s2. , vA/B = ?, aA/B = ?, W= 3 rev/min (constant)

X

Y

x

y

+t

vB

vBx

vBy

45°

vA

n

-t

0,)constant(/314.0seconds60

minute1

revolution1

radians2

minutesrevolution

3 W

W

srad

)/(627.0627.045sin45cos

/887.09

826.2

0

)/(22

45sin45cos

)/(826.2

)9(314.0

2

222

smjijaiaa

smR

vaa

Ra

smji

jvivv

sm

Rv

BBB

BBBn

Bt

BBB

B

W

WW

vA = 18 km/h, aA = 3 m/s2. , vA/B = ?, aA/B = ?, W= 3 rev/min (constant)

X

Y

x

y

+t

vB

vBx

vBy

45°

vA

n

-t

)/(3

)/(5

/5/18

2smia

smiv

smhkmv

A

A

A

W

smv

smjiv

vjii

vvv

BA

BA

BA

BABA

/61.3

)/(23

225

/

/

/

/

vB

vA

vA/B

vA = 18 km/h, aA = 3 m/s2. , vA/B = ?, aA/B = ?, W= 3 rev/min (constant)

X

x

y

aB

aBx

aBy

45°

+t

-t

n

W

aA

2

/

2

/

//

2

2

/68.3,)/(627.0627.3

627.0627.03,

)/(627.0627.0

)/(3

smasmjia

ajiiaaa

smjia

smia

BABA

BABABA

B

A

aB

aA

aA/B

2. Airplane A is flying horizontally with a constant speed of

200 km/h and is towing the glider B, which is gaining altitude.

If the tow cable has a length r = 60 m and q is increasing at

the constant rate of 5 degrees per second, determine the

magnitudes of the velocity and acceleration of the glider

for the instant when q = 15°. (2/196)

v

a

+r vB

aB=aBr vA

q

-r

-q

vAr

vAq

q

+q

re

qe

vA = 200 km/h (cst), r = 60 m, = 5 deg/s (cst), determine magnitudes of velocity and acceleration of glider for q = 15°.

q

ABAB

A

vvv

rr

srads

smhkmv

/

0

0

/087.05

/56.55/200

q

q

)()/(454.0

02,/454.0)087.0(60

,

0,

2

222

/

/

AtoBfromsmea

rrasmrra

eaeaaaa

aaaa

rB

BrB

BrrBBABB

AABAB

qqqq

qq

vB/A

vB

vA

vB/A

)/69.205(/14.57

)/(60.1967.5322.538.1467.53

,22.5

/22.5)087.0(60,

38.1467.5315sin56.5515cos56.55

/

///

hkmsmv

smeeeeev

evevvev

smrvevv

eeeev

B

rrB

BrBBAB

ABABAB

rrA

r

qqq

qq

q

qq

q

q

3. A batter hits the baseball A with an initial velocity of v 0 = 30 m/s directly toward fielder B at an angle of 30° to the horizontal; the initial position of the ball is 0.9 m above ground level. Fielder B requires ¼ s to judge where the ball should be caught and begins moving to that position with constant speed. Because of great experience, fielder B chooses his running speed so that he arrives at the “catch position” simultaneously with the baseball. The catch position is the field location at which the ball altitude is 2.1 m. Determine the velocity of the ball relative to the fielder at the instant the catch is made. (2/206)

v 0 = 30 m/s. Fielder B requires ¼ s to judge where the ball should be caught, then moves with constant speed, he arrives at catch position (y=2.1 m) simultaneously with the baseball. Determine velocity of the ball relative to fielder at the instant of catch.

mxxtvxx

ststtt

ttgttvyyBallA

x

y

44.12,)98.2(30cos30065

98.208.002.115905.4

905.430sin309.01.22

1

00

21

2

22

00

smvv

stttvxxPlayerB

BB

x

/55.4,)73.2(5644.1256

73.225.098.225.000

02 cbxax

v 0 = 30 m/s. Fielder B requires ¼ s to judge where the ball should be caught, then moves with constant speed, he arrives at catch position (y=2.1 m) simultaneously with the baseball. Determine velocity of the ball relative to fielder at the instant of catch.

)/(23.1443.2155.423.1498.25

55.4,/23.1498.25

/23.14)98.2(81.930sin30

/98.2530cos30

/

/

0

0

smjiijiv

ivsmjijvivv

smgtvv

smvv

vvv

BA

BAAA

AA

AA

BABA

yx

yy

xx

4. Particles A and B both have a

speed of 8 m/s along the directions

indicated by arrows. A moves in a

curvilinear path defined by y2 = x3

and B moves along a linear path

defined by y = -x. If the velocity of

B is decreasing at a rate of 6 m/s

each second and the velocity of A

is increasing at a rate of 5 m/s

each second, determine the

velocity and acceleration of A with

respect to B for the instant

represented.

vA=vB=8 m/s, velocity of B decreases at a rate of 6 m/s2, velocity

of A increases at a rate of 5 m/s 2, determine vA/B and aA/B.

vA

vB

5. Two particles A and B are moving at a speed of 4 m/s. Particle A travels along the spiral path r = 1.5q (m), where q is in radians, whereas particle B continues to move in a straight line. Particle A is decelerating at a rate of 2 m/s per second and radius of curvature of the path is 2.5 m when r = 1.5 m. At the same instant, particle B is accelerating at a rate of 5 m/s2. Determine the velocity and acceleration of particle A with respect to particle B in Cartesian coordinates at this instant.

B

A

r

q

r 1.5q

x

y

vA = vB = 4 m/s. Particle A has a spiral path with rA = 1.5q (m), q (rad). = -2 m/s2,

rA =2.5 m when r = 1.5 m. aB = 5 m/s2. Determine the velocity and acceleration of particle A with respect to particle B in Cartesian coordinates at this instant.

B

A

r

q

r 1.5q

x

y

Av

6. At the instant illustrated,

car B has a speed of 30 km/h

and is speeding up at a rate of

2.5m/s2. At the same instant

car A has a speed of 40 km/h

which is decreasing at a rate of

1.25 m/s2. Determine the

values of , , and for the

instant where r and q are

measured relative to a

longitudinal axis fixed to car B

as indicated in the figure.

r rq q

vB = 30 km/h, speeding up at a rate of 2.5m/s2. vA = 40 km/h, decreasing at a rate of 1.25 m/s2. Determine the values of , , and for the instant where r and q are measured relative to a longitudinal axis fixed to car B as indicated in the figure.

r rq q

b g g

g

+t

-t

n

+t

-t

n

+r

-r +q

+q

-q

-q