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A particle moves along a straight line a...

A particle moves along a straight line and its velocity depends on time as `v=6t-3t^(2)` where 'v' is in m/sec and 't' is in sec. Find average velocity and average speed for first four seconds.

Text Solution

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Average velocity `=("Total displacement")/("Total time")`
`v_(av) (underset(0)overset(4)int vdt)/(underset(0)overset(4)int dt) = (6 underset(0)overset(4)int t dt - 3 underset(0)overset(4)int t^(2) dt)/(4)`
`= (48 - 64)/(4) = - 4 m//s`
For average speed `= ("Total distance")/("Total time")`
`v = 6t - 3t^(2) = 3t (2 - t)`
i.e., `v ge 0` for `0 le t le 2`
and `v le 0` for `2 lt t le 4`
Thus, average speed `=(|underset(0)overset(2)int v dt| + |underset(2)overset(4)int v dt|)/(underset(0)overset(4)int dt)`
`= (|6 underset(0)overset(2)int t dt - 3 underset(0)overset(2)int t^(2) dt | + |6 underset(2)overset(4)int t dt - 3 underset(2)overset(4)int t^(2) dt|)/(4)`
`= (|4| + |-20|)/(4) = (24)/(4) = 6 m//s`
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