Question

The velocity-time graph of a particle in one-dimensional motion is shown in Fig:
Which of the following formulae are correct for describing the motion of the particle over the time-interval $t_1\text{to}{t}_2$:

$A$: $x\left(t_2\right)=x\left(t_1\right)+v\left(t_1\right)(t_2-t_1)+\left(1/2\right)a(t_2-t_1{)}^2$

$B$: $v\left(t_2\right)=v\left(t_1\right)+a(t_2-t_1)$

$C$: $v_{\text{average}}=\left(x\right(t_2)-x(t_1\left)\right)/(t_2-t_1)$

$D$: $v_{\text{average}}=\left(v\right(t_2)-v(t_1\left)/\right(t_2-t_1\left)\right)$

$E$: $x\left(t_2\right)=x\left(t_1\right)+v_{\text{average}}(t_2-t_1)+\left(1/2\right)a_{\text{average}}(t_2-t_1{)}^2$

$F$: $x\left(t_2\right)-x\left(t_1\right)=$area under the $v-t$ curve bounded by the $t-\text{axis}$ and the dotted line shown .

Answer 1

$A$:

Find that the relations are in the form of equation of motion or not.

With the help of the given curve, we can see that the slope is non-uniform, hence acceleration $\left(\overrightarrow{a}\right)$ is not uniform so equations of motion can not be used directly.

As $\left(A\right)$ is in the form of equation of motion so this is not the correct formulae describing the motion of the particle.

$x\left(t_2\right)=x\left(t_1\right)+v\left(t_1\right)(t_2-t_1)+\left(1/2\right)a(t_2-t_1{)}^2$
$B$:

Find that the relations are in the form of equation of motion or not.

With the help of the given curve, we can see that the slope is non-uniform, hence acceleration $\left(\overrightarrow{a}\right)$ is not uniform so equations of motion can not be used directly.

As $\left(B\right)$ is in the form of equation of motion so this is not the correct formulae describing the motion of the particle.

$v\left(t_2\right)=v\left(t_1\right)+a(t_2-t_1)$
$C$:

Find that the relations are in the form of equation of motion or not.
With the help of the given curve, we can see that the slope is non-uniform, hence acceleration $\left(\overrightarrow{a}\right)$ is not uniform.

$c$ is correct formula describing the motion of the particle.

$v_{\text{average}}=\left(x\right(t_2)-x(t_1\left)\right)/(t_2-t_1)$
$D$:

Find that the relations are in the form of equation of motion or not.
With the help of the given curve, we can see that the slope is non-uniform, hence acceleration $\left(\overrightarrow{a}\right)$ is not uniform.

$D$ is correct formula describing the motion of the particle.

$v_{\text{average}}=\left(v\right(t_2)-v(t_1\left)\right)/(t_2-t_1)$
$E$:

Find that the relations are in the form of equation of motion or not.
With the help of the given curve, we can see that the slope is non-uniform, hence acceleration $\left(\overrightarrow{a}\right)$ is not uniform so equations of motion can not be used directly.
As $\left(e\right)$ is in the form of equation of motion so this is not the correct formulae describing the motion of the particle.

$x\left(t_2\right)=x\left(t_1\right)+v_{\text{average}}(t_2-t_1)+\left(1/2\right)a_{\text{average}}(t_2-t_1{)}^2$
$F$:
Find that the relations are in the form of equation of motion or not.
With the help of the given curve, we can see that the slope is non-uniform, hence acceleration $\left(\overrightarrow{a}\right)$ is not uniform.

$\left(F\right)$ is correct formula describing the motion of the particle.

$x\left(t_2\right)-x\left(t_1\right)=$area under the $v-t$ curve bounded by the $t-\text{axis}$ and the dotted line shown .