For time `t=0` to `6s`
We can write equation for this motion, it is `y=mx` type line where m is the slope of the line.
Here, `a=((5-0))/((6-0))*timplies a=5/6t(ms^-2)` …(i)
`(For 0letlt6s)`
From `6s` to `12s` acceleration is constant, we can write equation for this motion, it is `y=const ant`.
Hence `a=5(ms^-2)` `(For 6letlt12s)`
For time `t=0` to `6s`
From Eq. (i) `a=(dv)/(dt)=5/6timpliesdv=5/6tdt`
Integrating both sides, we get `underset0oversetvintdv=5/6underset0oversettinttdt`
`[v]_0^v=5/6[t^2/2]_0^t` or `(v-0)=5/12(t^2-0)` or `v=5/12t^2`
Here velocity at `t=6s` is `v_(t=6s)=5/12xx6^2=15ms^-1`
For time `t=0` to `6s`
The acceleration is constant.
`a=5(ms^-2)` or `(dv)/(dt)=5` or `dv=5dt`
Integrating both sides, `underset15oversetvintdv=underset6oversettint5*dt`
`[v]_15^v=5[t]_6^t`
`v-15=5[t-6]`
`impliesv=5t-15`
Velocity-time relation will be a straight line of type `y=mx+c`.
Plotting velocity-time relations
For time `t=0` to `6s`
`v=5/12t^2`
Velocity time graph will be parabola passing through origin.
Position time relations
For time `t=0` to `6s`
As we know `v=(dx)/(dt)=5/12t^2`
Hence, `5/12t^2=(dx)/(dt)impliesdx=5/12int t^2dt`
Integrating both sides, `underset (0) oversetx int dx=5/12underset (0) overset (t) int t^2dt`
`(x-0)=5/12[t^3/3]_0^timpliesx=5/36t^3`
Position or the distance travelled from 0 to 6s
`x_((t=6sec))=5/36xx(6)^3=30m`
For time `t=6s` to `12s`
`v=5t-15implies(dx)/(dt)=5t-5`
or `dx=(5t-15)dt`
Integrating both sides
`underset(30)overset(x)intdx=underset6oversettint(5t-15)dt`
`[x]_30^x=5underset6oversettinttdt-15underset6oversettintdt`
`(x-30)=5[t^2/2]_6^t-15[t]_6^t`
`x-30=5/2[t^2-6^2]-15[t-6]`
`x=5/2t^2-15t+30`
At `t=12s`,
`x_((t=12s))=5/2(12)^2-15xx12+30=210m`