Question

A particle leaves the origin with an initial velocity $\vec{ u }=(3 \hat{ i }) \,ms ^{-1}$ and at a constant acceleration $\vec{ a }=(-1.0 \hat{ i }-0.5 \hat{ j }) \,ms ^{-2}$. Its velocity $v$ and position vector $\vec{ r }$ when it reaches its maximum $x$-coordinate are
(1) $v=-2 \hat{ j }\, ms ^{-1}$
(2) $v=(-1.5 \hat{ j }) \,ms ^{-1}$
(3) $\vec{ r }=(4.5 \hat{ i }-1.25 \hat{ j }) m$
(4) $\vec{ r }=(4.5 \hat{ i }-2.25 \hat{ j }) m$

• A. 1, 2 and 3 are correct
• B. 1 and 2 are correct
• C. 2 and 4 are correct
• D. 1 and 3 are correct

Answer 1

Answer: (C)

$u_{x}=3\, ms ^{-1}, a_{x}=-1.0\, ms ^{-2}$
$\therefore v_{x}^{2}=u_{x}^{2}+2 a_{x} x$
or $0=(3)^{2}+2(-1)(x)$
or $x=4.5\, m$
Also $v_{x}=u_{x}+a_{x} t$
$0=3-1.0\, t$
or $t=3 \,s$
$y=u_{y} t+\frac{1}{2} a_{y} t^{2}=0+\frac{1}{2}(-0.5)(3)^{2}=-2.25 \,m$
and $v_{y}=a_{y} t=(-0.5)(3)=-1.5\, ms ^{-1}$
$\therefore v=v_{x} \hat{ i }+v_{y} \hat{ j }=0-1.5 \hat{ j }=(-1.5 \hat{ j }) \,ms ^{-1}$
and $\vec{ r }=x \hat{ i }+\hat{y}=(4.5 \hat{ i }-2.25 \hat{ j })\, m$