Question

Derive : (Graphically)
$s_{n} = u + \dfrac{1}{2}a(2n - 1)$

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Answer 1

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$\red{\underline{{ \bf Question }}}$

• Derive : (Graphically)

$\red{\bf S_{n} = u + \dfrac{1}{2}a(2n - 1) }$

$\underline {\underline{{\purple{ \sf Solution }}}}$

The formula is about the displacement of a particle in n'th second

Distance Travelled (S) = Area of ▭ ABCD = Area of △ABE

$\sf S = ut + dfrac{1}{2}(v-u)$⠀⠀⠀⠀⠀— ❶

$\sf a = \dfrac{v-u}{t}$

$\sf at = v-u$⠀⠀⠀⠀⠀⠀— ❷

Therefore, ❷ in ❶

$\sf S = ut + \dfrac{1}{2} at^{2}$

$\tt S_{n} = [ u(n) + \dfrac{1}{2} a (n)^{2} ] - [u(n-1) + \dfrac{1}{2} a(n-1)^{2}]$

Therefore, $\tt { S_{n} = u + \dfrac{1}{2}a(2n - 1) }$

ᗩᗪᗪITIOᑎᗩᒪ IᑎᖴOᖇᗰᗩTIOᑎ

ʟᴀᴡ's ᴏғ ᴍᴏᴛɪᴏɴ

• ғɪʀsᴛ ʟᴀᴡ :- This law of Motion states that every object remains at rest or in motion until and unless an external force is applied. This law is also known as Law of Inertia.
• sᴇᴄᴏɴᴅ ʟᴀᴡ :- Second of law motion means that the momentum of object is proportional to applied unbalance force in the direction of force. This law gives us the derivation of Formula F ➡ m × a
• ᴛʜɪʀᴅ ʟᴀᴡ :- This law states that, for every reaction, there is an equal and opposite reaction.

ᴛʜʀᴇᴇ ᴇǫᴜᴀᴛɪᴏɴs ᴏғ ᴍᴏᴛɪᴏɴ

• v = u + αt
• s = ut + ½ αt²
• v² - u² = 2αs

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Answer 2

$\tt from \: 2nd \: eq. \: of \: motion \: we \: have$

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

$\tt here \: s = distance \: travelled$

$\tt u = initial \: velocity$

$\tt a = acceleraton$

$\tt t = time \: taken$

$\tt \small{distance \: travelled \: in \: n \: seconds \: is \: given \: by}$

$\tt s_{(n) } = un + \frac{1}{2} a {n}^{2} .....(i)$

$\tt \small{distance \: travelled \: in \: (n - 1) \: seconds \: is \: given \: by}$

$\tt s_{(n - 1) } = u(n - 1)+ \frac{1}{2} a {(n - 1)}^{2} .....(ii)$

$\tt s_{n } = s(n ) - s_{(n - 1) }$

$\tt \small{on \: putting \: the \: value \: of \: eq(i) \: and \: (ii) \: in \: above \: eq. \: we \: get}$

$\tt s_{n } = u + \frac{a}{2} (2n - 1)$