-The velocity of a particle is given by
v=2t²-3t+10 m/s. Find the instantaneous
acceleration at t=5s.
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-The velocity of a particle is given by
v=2t²-3t+10 m/s. Find the instantaneous
acceleration at t=5s.
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Given
Velocity of the particle is given by v = 2t² - 3t + 10
To Find
Instantaneous acceleration at t = 5 s
Knowledge Required
Acceleration at a particular time is called instantaneous acceleration
While solving these type of questions , we just need to differentiate the velocity .
Since , acceleration is defined as rate of change in velocity .
[tex]\bf \pink{\bigstar\ \; a=\dfrac{dv}{dt}}[/tex]
Solution
v = 2t² - 3t + 10
[tex]\implies \rm a=\dfrac{dv}{dt}\\\\\implies \rm a=\dfrac{d}{dt}(2t^2-3t+10)\\\\\implies \rm a=4t-3[/tex]
Now , we need to find instantaneous acceleration at t = 5 s .
[tex]\implies \rm a=4(5)-3\\\\\implies \rm a=20-3\\\\\implies \bf \green{a=17\ m/s^2\ \; \bigstar}[/tex]
Verified answer
[tex]\mathcal{\gray{\underline{\underline{\blue{GIVEN:-}}}}}[/tex]
[tex]\mathcal{\gray{\underline{\underline{\blue{TO\:FIND:-}}}}}[/tex]
[tex]\mathcal{\gray{\underline{\underline{\blue{SOLUTION:-}}}}}[/tex]
We have know that,
[tex]\green\bigstar\:\rm{\purple{\boxed{\red{Acceleration\:=\:\dfrac{Velocity}{Time}\:}}}}[/tex]
✍️ To find the instantaneous acceleration, we can calculate the differentiation of velocity function with respect to time .
[tex]\green\bigstar\:\rm{\purple{\boxed{\red{Instantaneous\:Acceleration\:=\:\dfrac{d(Velocity)}{d(Time)}\:}}}}[/tex]
[tex]\purple\bigstar\:\rm{\gray{\overbrace{\underbrace{\orange{Instantaneous\:acceleration\:=\:\dfrac{dv}{dt}\:}}}}}[/tex]
[tex]\rm{\implies\:Acceleration_{(insta)}\:=\:\dfrac{d(2t^2\:-\:3t\:+\:10)}{dt}\:}[/tex]
[tex]\rm{\implies\:Acceleration_{(insta)}\:=\:2\dfrac{d(t^2)}{dt}\:-\:3\dfrac{d(t)}{dt}\:+\:10\dfrac{d(1)}{dt}\:}[/tex]
[tex]\rm{\implies\:Acceleration_{(insta)}\:=\:2\times{2t}\:-\:3\times{1}\:+\:10\times{0}\:}[/tex]
[tex]\rm{\implies\:Acceleration_{(insta)}\:=\:4t\:-\:3\:+\:0\:}[/tex]
[tex]\rm{\green{\implies\:Acceleration_{(insta)}\:=\:4t\:-\:3}}[/tex]
⚡ Now put the value of “t = 5s” in the above equation
[tex]\rm{\implies\:Acceleration_{(insta)}\:=\:4\times{5}\:-\:3}[/tex]
[tex]\rm{\implies\:Acceleration_{(insta)}\:=\:20\:-\:3}[/tex]
[tex]\rm{\pink{\implies\:Acceleration_{(insta)}\:=\:17\:m/s^2\:}}[/tex]
[tex]\rm{\red{\therefore}}[/tex] The instantaneous acceleration at 5s is “17 m/s²” .