Name: Speedometer and Odometer
Uploaded: 2019-07-19
Duration: 4 min 16 s
Description: A speedometer is a gauge that measures and displays the instantaneous speed of a vehicle. An odometer is an instrument used for measuring the distance travel...

Question

Using the graphical method derive the second equation of motion

s = u t + \frac{1}{2} a t^{2}

Accepted Answer

Consider the velocity-time graph of an object that moves under uniform acceleration as depicted above. From this graph, you can see that initial velocity of the object is $u$ (at point A) and then it increases to $v$ (at point B) in time $t$ . The straight line graph represents that the body is under uniform accelerated motion with acceleration $a$ .

Now, consider the perpendicular lines $BC$ and $BE$ drawn from $B$ to time and velocity axis respectively. Now a little consideration will show that the $OA$ represents the initial velocity $(u)$ , $BC$ represents the final velocity $(v)$ and $OC$ represents the time interval $(t)$ . $AD$ is a line parallel to $OC$ .

From the graph,

$\begin{array}{rcl} BC & = & BD + DC = BD + OA \\ BC & = & BD + OA \\ v & = & BD + u \end{array} \begin{array}{rcl} \end{array} \begin{array}{rcl} \end{array}$

Therefore, $B D = v - u . . . . . . (1)$

We know that the acceleration $a = \frac{change in velocity}{time taken}$

From the graph, $a = \frac{BD}{AD} = \frac{BD}{OC}$

Substituting $OC = t$ , $BD = a \times t . . . . . . (2)$

We know that the distance travelled by the object is obtained by the area enclosed within $OABC$ under the velocity-time graph $AB$ .

Thus, the distance s travelled by the object is given by

$s =$ area $OABC$ $=$ area of the rectangle $OADC$ $+$ area of the triangle $ABD$

$s = OA \times OC + \frac{1}{2} (AD \times BD) . . . . . . (3)$

Substituting $OA = u$ , $OC = AD = t$ and $BD = a \times t$ in equation $(3)$

$s = u \times t + \frac{1}{2} \times (t \times a t)$

$s = u t + \frac{1}{2} a t^{2}$

$t$ -Time $(s)$	$0$	$10$	$20$	$30$	$40$	$50$
$s = u t + \frac{1}{2} a t^{2}$ - speed $(\frac{m}{s})$	$5$	$10$	$15$	$20$	$25$	$30$

Using the graphical method derive the second equation of motion $s = u t + \frac{1}{2} a t^{2}$

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Important Questions on Motion

Learn and Prepare for any exam you want !

Using the graphical method derive the second equation of motion $s = u t + \frac{1}{2} a t^{2}$