## Question:

A particle starts from s = 0 and travels along a straight line with a velocity $v = ({ t }^{ 2 } - 4t + 3) m /s$, where t is in seconds. Construct the v – t and a – t graphs for the time interval $0 \le t\le 4s$.

## Step-by-step

a-t Graph:

$a=\frac { dv }{ dt } =\frac { d }{ dt } ({ t }^{ 2 }-4t+3)\\a=(2t-4)m/{ s }^{ 2 }$

Thus,

$a{|}_{t=0}=2(0)-4=4m/{ s }^{ 2 }\\a{ | }_ { t=2 }=0\\a{ | }_{ t=4s }=2(4)-4=4m/{ s }^{ 2 }$

The at graph is shown in Fig. a.

v-t Graph: The slope of the v – t graph is zero when a $=\frac { dv }{ dt } =0$. Thus,

$a = 2t – 4 = 0 \quad\quad\quad t = 2s$

The velocity of the particle at t = 0 s, 2 s, and 4 s are

$v{ | }_{ t=0s } = { 0 }^{ 2 } – 4(0) + 3 = 3 m/s \\ v{ | }_{ t=2s } = { 2 }^{ 2 } – 4(2) + 3 = -1 m/s \\ v{ | }_{ t=4s } = { 4 }^{ 2 } – 4(4) + 3 = 3 m/s$

The vt graph is shown in Fig. b.