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V=(3t-9) find average speed and average velocity in 5sec
Asked by pramilac968 | 07 Jan, 2023, 11:02: PM
answered-by-expert Expert Answer
velocity v = dx/dt = (3t-9)
 
Position x as a function of time is determined by integrating above expression
 
x(t) = (3/2) t2 - 9t + C   .......................................(1)
 
where C is constant of integration or position x at t = 0
 
 
Average speed vavg  = distance / time
 
distance S travelled in 5 sec is determined using the following relation .
 
begin mathsize 14px style S space equals space integral subscript 0 superscript 5 space open parentheses square root of 1 plus open parentheses fraction numerator d x over denominator d t end fraction close parentheses squared end root close parentheses space d t space equals space integral subscript 0 superscript 5 square root of left parenthesis 3 t minus 9 right parenthesis squared plus 1 end root space d t end style
To perform above integration let us substitute u = ( 3 t - 9 ) , then above integration becomes
 
begin mathsize 14px style S space equals space 1 third integral subscript negative 9 end subscript superscript 6 square root of u squared plus 1 end root space d u space equals space 20.6 space m end style
Average speed = 20.6 m / 5 s = 4.12 m/s
 
------------------------------------------------------------------------------------------
 
Average velocity = displacement / time
 
displacement = x(5) - x(0)  = -7.5 m
 
( Eqn.(1) is used in above expression to get displacement )
 
Average velocity =  -7.5 m / 10 s = - 0.75 m/s
 
Answered by Thiyagarajan K | 08 Jan, 2023, 12:48: PM
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