Question

do it it is urgent. ​

Answer 1

✩ GIVEN ✩

• Tʜᴇ ᴀᴄᴄᴇʟᴇʀᴀᴛɪᴏɴ ᴏғ ᴀ ᴘᴀʀᴛɪᴄʟᴇ ɪs ɢɪᴠᴇɴ ʙʏ $\bf\red{a\:=\:3t^2\:+\:2t\:+\:2\:}$ [ᴡʜᴇʀᴇ, t ɪs ᴛʜᴇ ᴛɪᴍᴇ ɪɴ sᴇᴄᴏɴᴅ ] .

• $\bf\red{V_1}$ = 2 m/s at t = 0 s .

✩ TO FIND ✩

• $\bf\red{V_2}$ = ? at t = 2 s .

✩ SOLUTION ✩

☯︎ ᴡᴇ ʜᴀᴠᴇ ᴋɴᴏᴡ ᴛʜᴀᴛ, ᴛʜᴇ ʀᴀᴛᴇ ᴏғ ᴄʜᴀɴɢᴇ ᴏғ ᴠᴇʟᴏᴄɪᴛʏ ᴡɪᴛʜ ʀᴇsᴘᴇᴄᴛ ᴛᴏ ᴛɪᴍᴇ ɪs ᴋɴᴏᴡɴ ᴀs ᴀᴄᴄᴇʟᴇʀᴀᴛɪᴏɴ .

$\bf\pink{\dfrac{dv}{dt}\:=\:a\:}$

☯︎ Hᴇɴᴄᴇ, ᴛʜᴇ ɪɴᴛᴇɢʀᴀᴛɪᴏɴ ᴏғ ᴀᴄᴄᴇʟᴇʀᴀᴛɪᴏɴ-ᴛɪᴍᴇ ᴇǫᴜᴀᴛɪᴏɴ ɢɪᴠᴇs ᴛʜᴇ ᴠᴇʟᴏᴄɪᴛʏ .

$\bf{:\implies\:\displaystyle\int\limits_{v_1}^{v_2}{dv}\:=\:\displaystyle\int\limits_{t_1}^{t_2}{a\:.\:dt}\:}$

$\rm{:\implies\:\displaystyle\int\limits_{2}^{v_2}{dv}\:=\:\displaystyle\int\limits_{0}^{2}{(3t^2\:+\:2t\:+\:2)\:.\:dt}\:}$

$\rm{:\implies\:\Big[{v}\Big]_{2}^{v_2}\:=\:\Big[t^3\:+\:t^2\:+\:2t\:\Big]_{0}^{2}\:}$

• $\rm{:\implies\:v_2\:-\:2\:=\:2^3\:+\:2^2\:+\:2\times{2}\:}$

• $\rm{:\implies\:v_2\:-\:2\:=\:8\:+\:4\:+\:4\:}$

• $\rm{:\implies\:v_2\:-\:2\:=\:16\:}$

• $\rm{:\implies\:v_2\:=\:16\:+\:2\:}$

• $\bf\green{:\implies\:v_2\:=\:18\:m/s\:}$

$\huge\red\therefore$ Tʜᴇ ᴠᴇʟᴏᴄɪᴛʏ ᴀᴛ ᴛʜᴇ ᴇɴᴅ ᴏғ ᴛʜᴇ 2s ɪs "18m/s" .

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

__________•✩•____________________________________________________________________________

Answer 2

Answer:

✩ GIVEN ✩

Tʜᴇ ᴀᴄᴄᴇʟᴇʀᴀᴛɪᴏɴ ᴏғ ᴀ ᴘᴀʀᴛɪᴄʟᴇ ɪs ɢɪᴠᴇɴ ʙʏ $\bf\red{a\:=\:3t^2\:+\:2t\:+\:2\:}$ [ᴡʜᴇʀᴇ, t ɪs ᴛʜᴇ ᴛɪᴍᴇ ɪɴ sᴇᴄᴏɴᴅ ] .

$\bf\red{V_1}$ = 2 m/s at t = 0 s .

✩ TO FIND ✩

$\bf\red{V_2}$ = ? at t = 2 s .

✩ SOLUTION ✩

☯︎ ᴡᴇ ʜᴀᴠᴇ ᴋɴᴏᴡ ᴛʜᴀᴛ, ᴛʜᴇ ʀᴀᴛᴇ ᴏғ ᴄʜᴀɴɢᴇ ᴏғ ᴠᴇʟᴏᴄɪᴛʏ ᴡɪᴛʜ ʀᴇsᴘᴇᴄᴛ ᴛᴏ ᴛɪᴍᴇ ɪs ᴋɴᴏᴡɴ ᴀs ᴀᴄᴄᴇʟᴇʀᴀᴛɪᴏɴ .

$\bf\pink{\dfrac{dv}{dt}\:=\:a\:}$

☯︎ Hᴇɴᴄᴇ, ᴛʜᴇ ɪɴᴛᴇɢʀᴀᴛɪᴏɴ ᴏғ ᴀᴄᴄᴇʟᴇʀᴀᴛɪᴏɴ-ᴛɪᴍᴇ ᴇǫᴜᴀᴛɪᴏɴ ɢɪᴠᴇs ᴛʜᴇ ᴠᴇʟᴏᴄɪᴛʏ .

$\bf{:\implies\:\displaystyle\int\limits_{v_1}^{v_2}{dv}\:=\:\displaystyle\int\limits_{t_1}^{t_2}{a\:.\:dt}\:}$

$\rm{:\implies\:\displaystyle\int\limits_{2}^{v_2}{dv}\:=\:\displaystyle\int\limits_{0}^{2}{(3t^2\:+\:2t\:+\:2)\:.\:dt}\:}$

$\rm{:\implies\:\Big[{v}\Big]_{2}^{v_2}\:=\:\Big[t^3\:+\:t^2\:+\:2t\:\Big]_{0}^{2}\:}$

$\rm{:\implies\:v_2\:-\:2\:=\:2^3\:+\:2^2\:+\:2\times{2}\:}$

$\rm{:\implies\:v_2\:-\:2\:=\:8\:+\:4\:+\:4\:}$

$\rm{:\implies\:v_2\:-\:2\:=\:16\:}$

$\rm{:\implies\:v_2\:=\:16\:+\:2\:}$

$\bf\green{:\implies\:v_2\:=\:18\:m/s\:}$

$\huge\red\therefore$ Tʜᴇ ᴠᴇʟᴏᴄɪᴛʏ ᴀᴛ ᴛʜᴇ ᴇɴᴅ ᴏғ ᴛʜᴇ 2s ɪs "18m/s" .