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A particle travels so that its accelerat...

A particle travels so that its acceleration is given by `vec(a)=5 cos t hat(i)-3 sin t hat(j)`. If the particle is located at `(-3, 2)` at time `t=0` and is moving with a velocity given by `(-3hat(i)+2hat(j))`. Find
(i) The velocity `[vec(v)=int vec(a).dt]` at time t and
(ii) The position vector `[vec(r)=int vec(v).dt]` of the particle at time `t (t gt 0)`.

Text Solution

Verified by Experts

The correct Answer is:
(i) `vec(v)=(5sint-3)hat(i)+(3 cost-1)hat(j)`, (ii) `(2-5 cost-3t)hat(i)+(2+3sin t-t)hat(j)`
`vec(a)=5 cos that(i)-3 sin that(j)`
`rArr int dvec(v)=int 5 cos t dthat(i)-int 3 sin tdthat(j)`
Therefore `underset(-3)overset(v)(int)dv_(x)=underset(0)overset(t)(int)5 cos tdt rArr v_(x)=5 sin t-3`
`(dx)/(dt)=(5 sint-3)rArr underset(-3)overset(x)(int)dx=underset(0)overset(t)(int)(5 sint-3)dt`
`x+3=5-5 cost-3t rArr x=2-5 cost-3t`
Similarly,
`underset(2)overset(v)(int)dv_(y)=-underset(0)overset(t)(int)3 sin tdt`
`rArr v_(y)-2=3(cost-1)rArr v_(y)=3 cost-1`
`rArr underset(2)overset(y)(int)dy=underset(0)overset(t)(int)(3 cost-1)dt`
`rArr y-2=3 sin-t rArr y=2+3 sint-t`
Thus, `vec(v)=(5sint-3)hat(i)|(3cos t-1)hat(j)`
and `vec(s)=(2-5 cos t-3t)hat(i)+(2+3 sin t-t)hat(j)`
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