If velocity of a partical moving along a straight line changes sinusoidally with time as shown in given graph. Find the average speed over time interval `t = 0` to `t= 2 (2 n - 1)` second, `n` being any positive interget.

Answer

`(8)/(pi) m//s`

Explanation

Method1 :
Average speed `= ("Total distance travell ed")/("Total time taken") = (g)/(2 (2n - 1))`
Here `Delta t = 2 (2n - 1) = 4n - = 4 (n - 1) +2`
From the graph it is clear that time period `T is 4 s`
`:. Delta T = (n - 1) T + (T)/(2)`
Total distance travelled in one time period `= 4A` where `A` is amplitude.
Therefore, total distance travelled in `Delta t` is
`S = (n - 1) 4 A + 2A = (2n - 1) 2A`
`therefore gtvlt = ((2n - 1) 2A)/(2(2n - 1)) = A` and `omega A = v_(max) = 4`
`implies (2 pi)/(4) A = 4`
`implies A = (8)/(pi) :. gtvlt (8)/(pi) m//s`
Method 2
It can be observed from the graphthat average speed in time interval `t = 0` to `t = 2s` is same as that in intervals `t = 0` to `t = 4s` `t = 0`, to `t = 8s`, `t = 0` to `t = 12s`
or `t = 0` to `t = 2 (2n - 1)s`
The speed as function of time is
`v = |4 sin (2 pi)/(T) t| = |4 sin (2 pi)/(4) t| = |4 sin (pi t)/(2) `
`v = |4 sin ((2 pi)/(T)) t| = |4 sin ((2 pi)/(4)) t| = |4 sin ((pi t)/(2))|`
The average speed on time interval `t=0` to `t=2s` is
`overlinev=(int_0^2vdt)/(int_0^2dt)=(int_0^(2)4sin` `(pit)/(2)dt)/(2)=(8)/(pi)(m)/(s)`