The position of a particle moving along the x axis is given in centimeters by x

Question

The position of a particle moving along the x axis is given in centimeters by x = 6.25 + 1.50t^3, where t is in seconds. Consider the time interval t = 2.00 s to t = 3.00 s. Calculate the average velocity,_____________cm/s Calculate the instantaneous velocity at t = 2.00 s.___________cm/s Calculate the instantaneous velocity at t = 3.00 s.____________cm/s Calculate the instantaneous velocity at t = 2.50 s.____________cm/s Calculate the instantaneous velocity when the particle is midway between its positions at t = 2.00 s and t = 3.00 s.______________cm/s

Explanation / Answer

given

x = 6.25 + 1.5 t3  

a )

to get average velocity

t = 2 => x2 = 6.25 + 1.5 X 23

= 18.25 cm

t = 3 => x3 = 6.25 + 1.5 X 33

= 46.75 cm

dt = 3 - 2 = 1 sec

average velocity = 46.75 - 18.25 / 1

average velocity = 28.5 cm/sec

b )

t = 2 sec

x = 6.25 + 1.5 t3  

v =dx/dt

v = 4.5 t2

v = 4.5 X 4

v = 18 cm /sec

c )

t = 3 sec

x = 6.25 + 1.5 t3  

v =dx/dt

v = 4.5 t2

v = 4.5 X 9

v = 40.5 cm /sec

d )

t = 2.5 sec

x = 6.25 + 1.5 t3  

v =dx/dt

v = 4.5 t2

v = 4.5 X 2.52

v = 28.125 cm /sec

e )

xm = x2 + ( x3 - x2 ) / 2

x2 = 18.25 cm ; x3 = = 46.75 cm

xm = 18.25 + ( 46.75 - 18.25) / 2

xm = 32.5 cm

xm = 6.25 + 1.5 tm3  

32.5 = 6.25 + 1.5 tm3  

tm3 = 17.5

tm = 2.59 sec

vm = 4.5 tm2

vm = 4.5 X 2.592

vm = 30.18 cm/sec

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