Question

An object moves along x-axis such that its velocity varies with time as v = (50 - 10t) m/s, where t is time in seconds. Average speed of the particle in the time interval t = 0 to t = 6 s is equal to​

Answer 1

if t=0 then speed of particle = (50 - 10(0) )

= 50 - 0

= 50 m/s

t = 1

( 50 - 10(1) )

50 - 10

= 40 m/s

t= 2

( 50 - 10(2) )

( 50 - 20 )

= 30 m/s

t = 3

( 50 - 30 )

= 20 m/s

t = 4

= 10 m/s

t = 5

= 0 m/s

t = 6

= -10 m/s ( means that v of particle is in opposite directions in which it is moving before )

average speed = sum of speed / total time taken

= 50 + 40 + 30 + 20 + 10 +0 + -10 / 6

140/6

= 23.33 m/s is average speed

Answer 2

Given :  An object moves along x-axis such that its velocity varies with time as v = (50 - 10t) m/s

To Find :  Average speed of the particle in the time interval t = 0 to t = 6 s  

Solution:

v = (50 - 10t)

at t = 5   , v = 0

S = vt

 $S=\int\limits^5_0 {(50-10t)}t \, dt +\int\limits^6_5 {(50-10t)}t \, dt$

$S=\int\limits^5_0 {(50t-10t^2)} \, dt +\int\limits^6_5 {(50t-10t^2)} \, dt$

$S=\left [(25t^2-\frac{10t^3}{3} ) \right] _0^5 + \left [(25t^2-\frac{10t^3}{3} ) \right] _5^6$

S = 25(25) -  10(125)/3   +   25(36)  - 10(6)³/3  - 25(25)  +  10(125)/3

S = 180

Time = 6

Average speed  = 180/6  = 30  m/s

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