Question

A force f= 6t^2i +4tj is applied to a body of mass 4 kg at rest .what is the velocity of the body at end of 2s​

Answer 1

$\maltese\:\underline{\underline{\sf AnsWer :}}\:\maltese$

$\footnotesize\bullet\:\sf \overrightarrow{F} = 6t^2 \hat{i} + 4t \hat{j}$

$\footnotesize\bullet\:\sf m= 4 \: kg$

$\footnotesize\bullet\:\sf \overrightarrow{u} = 0 \: {ms}^{ - 1}$

CALCULATION :

$\longrightarrow\:\:\sf \overrightarrow{F} = m \overrightarrow{a}$

$\longrightarrow\:\:\sf\overrightarrow{a} = \dfrac{\overrightarrow{F}}{m}$

$\longrightarrow\:\:\sf\overrightarrow{a} = \dfrac{6t^2 \hat{i} + 4t \hat{j}}{4}$

$\longrightarrow\:\:\sf\overrightarrow{a} = \dfrac{6t^2 \hat{i} }{4} + \dfrac{ 4t \hat{j}}{4}$

$\longrightarrow\:\:\sf\overrightarrow{a} = \dfrac{3t^2 \hat{i} }{2} + t \hat{j}$

$\qquad \qquad \: \footnotesize\bullet\:\sf \overrightarrow{a}=\dfrac{\overrightarrow{dv}}{dt}$

$\longrightarrow\:\:\sf \dfrac{d\overrightarrow{v}}{dt} = \dfrac{3t^2 \hat{i} }{2} + t \hat{j}$

$\longrightarrow\:\:\sf d\overrightarrow{v}= \bigg (\dfrac{3t^2 \hat{i} }{2} + t \hat{j} \bigg)dt$

Integrating both the sides we have :

$\longrightarrow\:\:\displaystyle \sf \int\limits_{\overrightarrow{u}}^{\overrightarrow{v}}d\overrightarrow{v}= \int\limits_{t = 0}^{t = 2}\bigg (\dfrac{3t^2 \hat{i} }{2} + t \hat{j} \bigg)dt$

$\longrightarrow\:\:\displaystyle \sf \overrightarrow{v} - \overrightarrow{u}= \dfrac{3}{2} \int\limits_{t = 0}^{t = 2}t^2 \hat{i} \: dt+\int\limits_{t = 0}^{t = 2} t \hat{j} \: dt$

$\longrightarrow\:\:\displaystyle \sf \overrightarrow{v} - 0= \dfrac{3}{2} \Bigg[ \frac{t^{3} }{3} \Bigg]^2_0\hat{i}+\Bigg[ \dfrac{ {t}^{2} }{2} \Bigg] ^2_0\hat{j}$

$\longrightarrow\:\:\displaystyle \sf \overrightarrow{v} = \dfrac{1}{2} \Bigg[ t^{3} \Bigg]^2_0\hat{i}+\Bigg[ \dfrac{ {t}^{2} }{2} \Bigg] ^2_0\hat{j}$

$\longrightarrow\:\:\displaystyle \sf \overrightarrow{v} = \dfrac{1}{2} \Bigg[ (2)^{3} - {(0)}^{3} \Bigg]\hat{i}+\Bigg[ \dfrac{ {(2)}^{2} }{2} - \dfrac{ {0}^{2} }{2} \Bigg] \hat{j}$

$\longrightarrow\:\:\displaystyle \sf \overrightarrow{v} = \dfrac{1}{2} \Bigg[ 8 \Bigg]\hat{i}+\Bigg[ \dfrac{ 4}{2} \Bigg] \hat{j}$

$\longrightarrow\:\:\displaystyle \sf \overrightarrow{v} = \dfrac{8}{2} \hat{i}+ \dfrac{ 4}{2} \hat{j}$

$\longrightarrow\: \: \underline{ \underline{\displaystyle \sf \overrightarrow{v} = 4 \hat{i}+ 2 \hat{j}}}$