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Class 10 Class 12

The position of a particle is given by
                space straight r with rightwards arrow on top equals 3 straight t straight i with hat on top minus 2 straight t squared straight j with hat on top plus 4 straight k with hat on top
where t is in second and r is in meter.
(a) Find the straight v with rightwards arrow on top space and space straight a with rightwards arrow on top of the particle.
(b) What is the magnitude and direction of velocity of the particle at t = 2s?


(a)

Given,

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Velocity,

space space straight v with rightwards arrow on top equals straight d over dt left parenthesis straight r with rightwards arrow on top right parenthesis
space space space space equals straight d over dt left parenthesis 3 straight t straight i with hat on top minus 2 straight t squared straight j with hat on top plus 4 straight k with hat on top right parenthesis

and,

Acceleration,space space space straight a with rightwards arrow on top equals straight d over dt left parenthesis straight v with rightwards arrow on top right parenthesis space equals straight d over dt left parenthesis 3 straight i with hat on top minus 4 straight t straight j with hat on top right parenthesis
                        equals negative 4 straight j with hat on top

(b) Now,

Velocity,space space straight v with rightwards arrow on top space equals space 3 straight i with hat on top minus 4 straight t straight j with hat on top

At time = 2 sec, 

           straight v with rightwards arrow on top space equals space 3 straight i with hat on top minus 8 straight j with hat on top

∴  open vertical bar straight v with rightwards arrow on top close vertical bar space equals space square root of left parenthesis 3 right parenthesis squared plus left parenthesis 8 right parenthesis squared end root space equals space square root of 74 space equals space 8.54 space straight m divided by straight s

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At t = 2 sec,

Acceleration, straight a with rightwards arrow on top space equals space minus 4 straight j with hat on top

∴        open vertical bar straight a with rightwards arrow on top close vertical bar space equals space 4 and directed towards negative direction of X-axis. 

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