Question

obtain equations of motion for constant acceleration using method of calculus​

Answer 1

please mark as BRAINLIEST ❤️❤️❤️

Answer 2

ʜɪɪ☺️

ǫᴜᴇsᴛɪᴏɴ

ᴏʙᴛᴀɪɴ ᴇǫᴜᴀᴛɪᴏɴs ᴏғ ᴍᴏᴛɪᴏɴ ғᴏʀ ᴄᴏɴsᴛᴀɴᴛ ᴀᴄᴄᴇʟᴇʀᴀᴛɪᴏɴ ᴜsɪɴɢ ᴍᴇᴛʜᴏᴅ ᴏғ ᴄᴀʟᴄᴜʟᴜs

ᴀɴsᴡᴇʀ:-

ᴀᴄᴄᴇʟᴇʀᴀᴛɪᴏɴ ᴀ ɪs ᴄᴏɴsᴛᴀɴᴛ.

ᴡᴇ ᴋɴᴏᴡ-

ᴅ2s /ᴅᴛ2 =ᴅᴠ/ᴅᴛ =ᴀ

ɪɴᴛᴇɢʀᴀᴛɪɴɢ ᴀ ᴡɪᴛʜ ʀᴇsᴘᴇᴄᴛ ᴛᴏ ᴛɪᴍᴇ ᴡɪʟʟ ʟᴇᴀᴠᴇ ᴜs ᴡɪᴛʜ ᴀɴ ᴇxᴘʀᴇssɪᴏɴ ғᴏʀ ᴠᴇʟᴏᴄɪᴛʏ, ᴠ , (ᴀs ᴀᴄᴄᴇʟᴇʀᴀᴛɪᴏɴ ɪs ᴛʜᴇ ʀᴀᴛᴇ ᴏғ ᴄʜᴀɴɢᴇ ᴏғ ᴠᴇʟᴏᴄɪᴛʏ, ᴡʀᴛ ᴛɪᴍᴇ):

ᴠ=ᴅs/ᴅᴛ = ∫ᴀ ᴅᴛ = ᴀᴛ+ᴜ,

ᴠ= ᴜ+ᴀᴛ .......(1)

ᴡʜᴇʀᴇ ᴜ ɪs ᴀ ᴄᴏɴsᴛᴀɴᴛ ᴏғ ɪɴᴛᴇɢʀᴀᴛɪᴏɴ. ɪɴ ᴄᴏɴᴛᴇxᴛ ɪᴛ ɪs ᴛʜᴇ ɪɴɪᴛɪᴀʟ ᴠᴇʟᴏᴄɪᴛʏ ᴀ ʙᴏᴅʏ ɪs ᴛʀᴀᴠᴇʟʟɪɴɢ ʙᴇғᴏʀᴇ ɪᴛ ᴀᴄᴄᴇʟᴇʀᴀᴛᴇs.

ɪɴᴛᴇɢʀᴀᴛɪɴɢ ᴠᴇʟᴏᴄɪᴛʏ ᴡʀᴛ ᴛɪᴍᴇ ᴡɪʟʟ ʟᴇᴀᴠᴇ ᴜs ᴡɪᴛʜ ᴀɴ ᴇxᴘʀᴇssɪᴏɴ ғᴏʀ ᴅɪsᴘʟᴀᴄᴇᴍᴇɴᴛ (ᴀs ᴠᴇʟᴏᴄɪᴛʏ ɪs ᴛʜᴇ ғɪʀsᴛ ᴅᴇʀɪᴠᴀᴛɪᴠᴇ ᴏғ ᴅɪsᴘʟᴀᴄᴇᴍᴇɴᴛ ᴍᴏᴅᴇʟʟᴇᴅ ᴀs ᴀ ғᴜɴᴄᴛɪᴏɴ ᴏғ ᴛɪᴍᴇ):

s =∫(ᴅs/ᴅᴛ) ᴅᴛ = ∫(ᴀᴛ+ᴜ) ᴅᴛ = ∫ᴀᴛ ᴅᴛ + ∫ᴜ ᴅᴛ

ʜᴇɴᴄᴇ, s= ᴜᴛ+(1/2)ᴀᴛ2 ...........(2)

ɴᴏᴛᴇ ᴛʜᴀᴛ ᴛʜɪs ᴇǫᴜᴀᴛɪᴏɴ ᴀssᴜᴍᴇs ᴅɪsᴘʟᴀᴄᴇᴍɴᴛ ɪs ᴢᴇʀᴏ ᴡʜᴇɴ ᴛɪᴍᴇ ᴛ= 0, ɪ.ᴇ. ᴛʜᴇ ᴄᴏɴsᴛᴀɴᴛ ᴏғ ɪɴᴛᴇɢʀᴀᴛɪᴏɴ ʜᴀs ʙᴇᴇɴ ɴᴇɢʟᴇᴄᴛᴇᴅ.

ғʀᴏᴍ ᴇǫᴜᴀᴛɪᴏɴ (2)

s = 1/2(2ᴜ+ᴀᴛ) ᴛ

s= 1/2(ᴜ+ᴜ+ᴀᴛ)ᴛ = 1/2(ᴜ+ᴠ)ᴛ

ᴠ2 = (ᴜ+ᴀᴛ)2 =ᴜ2 +2ᴜᴀᴛ+ᴀᴛ2

ᴠ2 = ᴜ2 +2ᴀ(ᴜᴛ+1/2 ᴀᴛ2)

= ᴜ2+2ᴀs

ᴠ2=ᴜ2+2ᴀs .............(3)

ᴡᴇ ᴋɴᴏᴡ, ᴀᴄᴄᴇʟᴇʀᴀᴛɪᴏɴ ɪs ᴛʜᴇ (ᴄʜᴀɴɢᴇ ɪɴ ᴠᴇʟᴏᴄɪᴛʏ / ᴜɴɪᴛ ᴛɪᴍᴇ).

ᴄᴏɴsɪᴅᴇʀ ᴀɴ ɪɴsᴛᴀɴᴛ ᴛɪᴍᴇ; ᴀ = (ᴅᴠ/ᴅᴛ)

ᴀ = (ᴅᴠ/ᴅᴛ)

ᴀ ∫ᴅᴛ = ∫ᴅᴠ

ɪɴᴛᴇɢʀᴀᴛɪɴɢ

ᴀ.(ᴛ₂ - ᴛ₁) = (ᴠ – ᴜ)

ʟᴇᴛ (ᴛ₂- ᴛ₁) = ᴛ

ᴛʜᴇɴ, ᴀᴛ = ᴠ - ᴜ

ᴠ = ᴜ + ᴀᴛ ɪᴛs ᴀ ᴍᴏᴛɪᴏɴ ᴇǫᴜᴀᴛɪᴏɴ ,

ɴᴏᴡ ,

ᴠ = ᴜ + ᴀᴛ

ᴍᴜʟᴛɪᴘʟʏ ʙᴏᴛʜ sɪᴅᴇs, ᴡɪᴛʜ ᴅᴛ

ᴠ x ᴅᴛ = ᴜ x ᴅᴛ + ᴀᴛ xᴅᴛ

ᴡᴇ ᴋɴᴏᴡ, ᴠ x ᴅᴛ = s { ᴅɪsᴛᴀɴᴄᴇ = sᴘᴇᴇᴅ x ᴛɪᴍᴇ }

ᴜsɪɴɢ ᴛʜᴇ ᴀʙᴏᴠᴇ ᴇǫᴜᴀᴛɪᴏɴ

s = ᴜ(ᴛ₂-ᴛ₁) + ᴀ(ᴛ₂-ᴛ₁)²/(2)

ᴛʜᴇʀᴇғᴏʀᴇ,

s = ᴜᴛ + (1/2)ᴀᴛ² ᴍᴏᴛɪᴏɴ, ᴇǫᴜᴀᴛɪᴏɴ.

ᴀʟsᴏ

s = ᴜᴛ + (1/2)ᴀᴛ² _(1)

ᴠ = ᴜ + ᴀᴛ

sǫᴜᴀʀɪɴɢ ʙᴏᴛʜ sɪᴅᴇs,

ᴠ² = ᴜ² + ᴀ²ᴛ² + 2ᴜᴀᴛ

ᴠ² = ᴜ² + 2ᴀ {(1/2) ᴀᴛ² + ᴜᴛ}

ғʀᴏᴍ ᴇǫᴜ (1)

ᴠ² = ᴜ² + 2ᴀs ᴍᴏᴛɪᴏɴ ᴇǫᴜᴀᴛɪᴏɴ,

ᴏʀ ᴡᴇ ᴋɴᴏᴡ,

ᴠx(ᴅᴠ/ᴅx) = ᴀ

∫ᴠ ᴅᴠ = ᴀ ∫ᴅx

ɪɴᴛᴇɢʀᴀᴛᴇ ʙᴏᴛʜ sɪᴅᴇs,

(ᴠ² - ᴜ²) = 2ᴀ(x₂ - x₁)

ʟᴇᴛ (x₂ - x₁) = s

ᴠ² = ᴜ² + 2ᴀs ᴛʜɪʀᴅ ᴇǫᴜᴀᴛɪᴏɴ ᴏғ ᴍᴏᴛɪᴏɴ